College Algebra Unit 3 Polynomial and Rational Functions Review – Step-by-Step Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. For the polynomial "},{"type":"inlineMath","attrs":{"latex":"f(x) = 6x^5 - 12x^4 + 20x^3 - 7x + 1"}},{"type":"text","marks":[{"type":"bold"}],"text":":"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(a) Find the degree of the polynomial."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(b) Find the leading term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(c) Find the leading coefficient."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Polynomial Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the structure of polynomials, specifically how to identify the degree, leading term, and leading coefficient."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Degree:"},{"type":"text","text":" The highest power of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in the polynomial."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Leading Term:"},{"type":"text","text":" The term with the highest degree."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Leading Coefficient:"},{"type":"text","text":" The coefficient of the leading term."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Look at all the terms in "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" and identify the one with the highest exponent of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The degree is the exponent of this term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The leading term is the entire term with the highest degree (including its coefficient and variable)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The leading coefficient is the number multiplying the variable in the leading term."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(a) Degree: 5"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(b) Leading term: "},{"type":"inlineMath","attrs":{"latex":"6x^5"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(c) Leading coefficient: 6"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The degree is determined by the highest exponent, the leading term is the term with that exponent, and the leading coefficient is its coefficient."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find the zeros of the polynomial function "},{"type":"inlineMath","attrs":{"latex":"g(x) = (x + 1)(x - 4)(x + 7)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and state the multiplicity of each."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Zeros of Polynomial Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the zeros (roots) of a polynomial written in factored form and to determine the multiplicity of each zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero (Root):"},{"type":"text","text":" A value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" for which "},{"type":"inlineMath","attrs":{"latex":"g(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Multiplicity:"},{"type":"text","text":" The number of times a particular zero occurs (the exponent on the factor)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero: "},{"type":"inlineMath","attrs":{"latex":"x + 1 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x - 4 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x + 7 = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve each equation for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the zeros."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check the exponents on each factor to determine the multiplicity (if no exponent is written, multiplicity is 1)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Zeros: "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":" (multiplicity 1), "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":" (multiplicity 1), "},{"type":"inlineMath","attrs":{"latex":"x = -7"}},{"type":"text","text":" (multiplicity 1)"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each factor is linear and appears only once, so each zero has multiplicity 1."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Find the zeros and their multiplicities for "},{"type":"inlineMath","attrs":{"latex":"h(x) = 2(x - 4)^3(x + 4)^2(x + 6)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Zeros and Multiplicities of Polynomial Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find zeros from a factored polynomial and to determine their multiplicities based on exponents."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero:"},{"type":"text","text":" Value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that makes "},{"type":"inlineMath","attrs":{"latex":"h(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Multiplicity:"},{"type":"text","text":" The exponent on the factor corresponding to each zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero: "},{"type":"inlineMath","attrs":{"latex":"x - 4 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x + 4 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x + 6 = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find each zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Read the exponent on each factor to determine the multiplicity: "},{"type":"inlineMath","attrs":{"latex":"(x - 4)^3"}},{"type":"text","text":" means multiplicity 3, etc."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":" (multiplicity 3)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = -4"}},{"type":"text","text":" (multiplicity 2)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = -6"}},{"type":"text","text":" (multiplicity 1)"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The exponents on each factor indicate the multiplicity of each zero."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Find the zeros and their multiplicities for "},{"type":"inlineMath","attrs":{"latex":"h(x) = 2x^3 - 2x^2 - x"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Factoring and Finding Zeros of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to factor a cubic polynomial and find its zeros and their multiplicities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Factoring:"},{"type":"text","text":" Rewriting the polynomial as a product of simpler polynomials."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero:"},{"type":"text","text":" Value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that makes "},{"type":"inlineMath","attrs":{"latex":"h(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Multiplicity:"},{"type":"text","text":" The number of times a zero occurs."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","text":" by first factoring out the greatest common factor."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check the exponents on each factor to determine multiplicity."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" (multiplicity 1)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","text":" (multiplicity 1)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = - rac{1}{2}"}},{"type":"text","text":" (multiplicity 1)"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"All zeros are simple (multiplicity 1) after factoring completely."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Find the zeros and their multiplicities for "},{"type":"inlineMath","attrs":{"latex":"h(x) = 3x^3 + 27x^2 - 9x"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Factoring and Finding Zeros of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to factor a cubic polynomial and find its zeros and their multiplicities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Factoring:"},{"type":"text","text":" Rewriting the polynomial as a product of simpler polynomials."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero:"},{"type":"text","text":" Value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that makes "},{"type":"inlineMath","attrs":{"latex":"h(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Multiplicity:"},{"type":"text","text":" The number of times a zero occurs."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor out the greatest common factor from all terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor the remaining quadratic if possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check the exponents on each factor to determine multiplicity."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" (multiplicity 1)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = -9"}},{"type":"text","text":" (multiplicity 1)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","text":" (multiplicity 1)"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"All zeros are simple (multiplicity 1) after factoring completely."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Use synthetic division to find the quotient and remainder of "},{"type":"inlineMath","attrs":{"latex":"x^3 - 7x^2 + 13x + 3 \\div (x - 2)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Synthetic Division"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use synthetic division to divide a cubic polynomial by a linear factor and find the quotient and remainder."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Synthetic Division:"},{"type":"text","text":" A shortcut method for dividing a polynomial by a linear factor of the form "},{"type":"inlineMath","attrs":{"latex":"(x - c)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Quotient:"},{"type":"text","text":" The result of the division (excluding the remainder)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Remainder:"},{"type":"text","text":" The leftover value after division."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up synthetic division using "},{"type":"inlineMath","attrs":{"latex":"c = 2"}},{"type":"text","text":" (from "},{"type":"inlineMath","attrs":{"latex":"x - 2"}},{"type":"text","text":") and the coefficients: "},{"type":"inlineMath","attrs":{"latex":"1, -7, 13, 3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Carry down the leading coefficient, multiply by "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":", and add to the next coefficient. Repeat for all coefficients."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The last number is the remainder; the other numbers are the coefficients of the quotient."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the quotient as a polynomial of degree one less than the original."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 5x + 3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remainder: $9$"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The synthetic division process yields these results."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Use synthetic division to find the quotient and remainder of "},{"type":"inlineMath","attrs":{"latex":"x^4 - 36 \\div (x + 3)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Synthetic Division"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use synthetic division to divide a quartic polynomial by a linear factor and find the quotient and remainder."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Synthetic Division:"},{"type":"text","text":" Used for dividing by "},{"type":"inlineMath","attrs":{"latex":"(x - c)"}},{"type":"text","text":"; here, "},{"type":"inlineMath","attrs":{"latex":"c = -3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Quotient:"},{"type":"text","text":" The result of the division (excluding the remainder)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Remainder:"},{"type":"text","text":" The leftover value after division."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the coefficients of "},{"type":"inlineMath","attrs":{"latex":"x^4 - 36"}},{"type":"text","text":" (include zeros for missing terms): "},{"type":"inlineMath","attrs":{"latex":"1, 0, 0, 0, -36"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up synthetic division with "},{"type":"inlineMath","attrs":{"latex":"c = -3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Carry down the leading coefficient, multiply by "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":", and add to the next coefficient, repeating for all coefficients."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The last number is the remainder; the others are the coefficients of the quotient."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient: "},{"type":"inlineMath","attrs":{"latex":"x^3 - 3x^2 + 9x - 27"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remainder: $45$"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The synthetic division process yields these results."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Using synthetic division, determine whether "},{"type":"inlineMath","attrs":{"latex":"-3"}},{"type":"text","marks":[{"type":"bold"}],"text":" and $2"},{"type":"inlineMath","attrs":{"latex":" are zeros of "}},{"type":"text","marks":[{"type":"bold"}],"text":"f(x) = 3x^3 + 5x^2 - 6x + 18$."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Zeros of Polynomials and Synthetic Division"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use synthetic division to check if a given value is a zero of a polynomial (i.e., if the remainder is zero)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero of a Polynomial:"},{"type":"text","text":" A value "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"f(c) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Synthetic Division:"},{"type":"text","text":" Used to test if a value is a zero by checking if the remainder is zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each value ("},{"type":"inlineMath","attrs":{"latex":"-3"}},{"type":"text","text":" and $2"},{"type":"inlineMath","attrs":{"latex":"), set up synthetic division with the coefficients $3, 5, -6, 18"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perform synthetic division for "},{"type":"inlineMath","attrs":{"latex":"c = -3"}},{"type":"text","text":" and record the remainder."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Repeat for "},{"type":"inlineMath","attrs":{"latex":"c = 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If the remainder is zero, the value is a zero of the polynomial."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-3"}},{"type":"text","text":" is a zero (remainder is 0)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"$2$ is not a zero (remainder is not 0)."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Synthetic division confirms which values are zeros."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Using synthetic division, determine whether "},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"-2"}},{"type":"text","marks":[{"type":"bold"}],"text":" are zeros of "},{"type":"inlineMath","attrs":{"latex":"f(x) = 4x^4 - 3x^3 + 14x^2 - 8x"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Zeros of Polynomials and Synthetic Division"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use synthetic division to check if a given value is a zero of a polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero of a Polynomial:"},{"type":"text","text":" A value "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"f(c) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Synthetic Division:"},{"type":"text","text":" Used to test if a value is a zero by checking if the remainder is zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each value ("},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"-2"}},{"type":"text","text":"), set up synthetic division with the coefficients "},{"type":"inlineMath","attrs":{"latex":"4, -3, 14, -8, 0"}},{"type":"text","text":" (add 0 for the constant term if missing)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perform synthetic division for "},{"type":"inlineMath","attrs":{"latex":"c = -1"}},{"type":"text","text":" and record the remainder."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Repeat for "},{"type":"inlineMath","attrs":{"latex":"c = -2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If the remainder is zero, the value is a zero of the polynomial."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","text":" is not a zero (remainder is not 0)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-2"}},{"type":"text","text":" is a zero (remainder is 0)."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Synthetic division confirms which values are zeros."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Find a polynomial function of degree 5 with "},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","marks":[{"type":"bold"}],"text":" as a zero of multiplicity 3, and $0$ as a zero of multiplicity 2."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Constructing Polynomials from Zeros and Multiplicities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to write a polynomial given its zeros and their multiplicities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Multiplicity:"},{"type":"text","text":" The exponent on the factor corresponding to each zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"General Form:"},{"type":"text","text":" If "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" is a zero of multiplicity "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":", the factor is "},{"type":"inlineMath","attrs":{"latex":"(x - c)^m"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write a factor for each zero: "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":" (multiplicity 3) gives "},{"type":"inlineMath","attrs":{"latex":"(x + 1)^3"}},{"type":"text","text":"; "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" (multiplicity 2) gives "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the factors together to get the polynomial."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Optionally, expand the product if required."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"f(x) = x^2(x + 1)^3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This polynomial has degree 5, with the required zeros and multiplicities."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Find a polynomial function of degree 3 with zeros: $2"},{"type":"inlineMath","attrs":{"latex":" (multiplicity 1), "}},{"type":"text","marks":[{"type":"bold"}],"text":"-1"},{"type":"inlineMath","attrs":{"latex":" (multiplicity 1), and $0"}},{"type":"text","marks":[{"type":"bold"}],"text":" (multiplicity 1)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Constructing Polynomials from Zeros"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to write a polynomial given its zeros and their multiplicities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Multiplicity:"},{"type":"text","text":" The exponent on the factor corresponding to each zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"General Form:"},{"type":"text","text":" If "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" is a zero of multiplicity "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":", the factor is "},{"type":"inlineMath","attrs":{"latex":"(x - c)^m"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write a factor for each zero: "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"(x - 2)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"(x + 1)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the factors together to get the polynomial."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Optionally, expand the product if required."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"f(x) = x(x - 2)(x + 1)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This polynomial has degree 3, with the required zeros and multiplicities."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. Use synthetic division to find all zeros of "},{"type":"inlineMath","attrs":{"latex":"f(x) = x^3 - 3x^2 - 3x + 3"}},{"type":"text","marks":[{"type":"bold"}],"text":" given that $1$ is a zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Synthetic Division and Finding All Zeros"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use synthetic division to reduce the degree of a polynomial and then find all zeros."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Synthetic Division:"},{"type":"text","text":" Used to divide by "},{"type":"inlineMath","attrs":{"latex":"(x - 1)"}},{"type":"text","text":" since $1$ is a zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Quadratic Factor:"},{"type":"text","text":" After division, factor the resulting quadratic to find remaining zeros."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up synthetic division with "},{"type":"inlineMath","attrs":{"latex":"c = 1"}},{"type":"text","text":" and coefficients "},{"type":"inlineMath","attrs":{"latex":"1, -3, -3, 3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perform synthetic division to obtain the depressed quadratic."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the quadratic equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (factor or use quadratic formula)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Zeros: "},{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 1 + \\\\sqrt{2}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 1 - \\\\sqrt{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After synthetic division, the quadratic factors yield the remaining zeros."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. Use synthetic division to find all zeros of "},{"type":"inlineMath","attrs":{"latex":"f(x) = x^3 - 3x + 2"}},{"type":"text","marks":[{"type":"bold"}],"text":" given that "},{"type":"inlineMath","attrs":{"latex":"-2"}},{"type":"text","marks":[{"type":"bold"}],"text":" is a zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Synthetic Division and Finding All Zeros"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use synthetic division to reduce the degree of a polynomial and then find all zeros."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Synthetic Division:"},{"type":"text","text":" Used to divide by "},{"type":"inlineMath","attrs":{"latex":"(x + 2)"}},{"type":"text","text":" since "},{"type":"inlineMath","attrs":{"latex":"-2"}},{"type":"text","text":" is a zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Quadratic Factor:"},{"type":"text","text":" After division, factor the resulting quadratic to find remaining zeros."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up synthetic division with "},{"type":"inlineMath","attrs":{"latex":"c = -2"}},{"type":"text","text":" and coefficients "},{"type":"inlineMath","attrs":{"latex":"1, 0, -3, 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perform synthetic division to obtain the depressed quadratic."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the quadratic equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (factor or use quadratic formula)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Zeros: "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = -1"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After synthetic division, the quadratic factors yield the remaining zeros."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. List all possible rational zeros for "},{"type":"inlineMath","attrs":{"latex":"h(x) = 3x^4 - 5x^3 + x^2 - x + 1"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Zero Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the Rational Zero Theorem to list all possible rational zeros of a polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Rational Zero Theorem:"},{"type":"text","text":" Possible rational zeros are "},{"type":"inlineMath","attrs":{"latex":"\\pm \\frac{p}{q}"}},{"type":"text","text":", where "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":" divides the constant term and "},{"type":"inlineMath","attrs":{"latex":"q"}},{"type":"text","text":" divides the leading coefficient."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the constant term ("},{"type":"inlineMath","attrs":{"latex":"a_0"}},{"type":"text","text":") and leading coefficient ("},{"type":"inlineMath","attrs":{"latex":"a_n"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List all factors of the constant term (numerators "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List all factors of the leading coefficient (denominators "},{"type":"inlineMath","attrs":{"latex":"q"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Form all possible fractions "},{"type":"inlineMath","attrs":{"latex":"\\pm \\frac{p}{q}"}},{"type":"text","text":" and list them (simplify if possible)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Possible rational zeros: "},{"type":"inlineMath","attrs":{"latex":"\\pm1, \\pm\\frac{1}{3}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are all combinations of factors of 1 (constant) over factors of 3 (leading coefficient)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. List all possible rational zeros for "},{"type":"inlineMath","attrs":{"latex":"h(x) = 6x^3 - 2x^2 + 7x - 18"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Zero Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the Rational Zero Theorem to list all possible rational zeros of a polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Rational Zero Theorem:"},{"type":"text","text":" Possible rational zeros are "},{"type":"inlineMath","attrs":{"latex":"\\pm \\frac{p}{q}"}},{"type":"text","text":", where "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":" divides the constant term and "},{"type":"inlineMath","attrs":{"latex":"q"}},{"type":"text","text":" divides the leading coefficient."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the constant term ("},{"type":"inlineMath","attrs":{"latex":"a_0"}},{"type":"text","text":") and leading coefficient ("},{"type":"inlineMath","attrs":{"latex":"a_n"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List all factors of the constant term (numerators "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List all factors of the leading coefficient (denominators "},{"type":"inlineMath","attrs":{"latex":"q"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Form all possible fractions "},{"type":"inlineMath","attrs":{"latex":"\\pm \\frac{p}{q}"}},{"type":"text","text":" and list them (simplify if possible)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Possible rational zeros: "},{"type":"inlineMath","attrs":{"latex":"\\pm1, \\pm2, \\pm3, \\pm6, \\pm9, \\pm18, \\pm\\frac{1}{2}, \\pm\\frac{3}{2}, \\pm\\frac{9}{2}, \\pm\\frac{1}{3}, \\pm\\frac{2}{3}, \\pm\\frac{6}{3}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are all combinations of factors of "},{"type":"inlineMath","attrs":{"latex":"-18"}},{"type":"text","text":" (constant) over factors of $6$ (leading coefficient), simplified."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Find all the asymptotes of "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x^2 + 5x + 4}{x^2 + x - 32}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and label them appropriately."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Functions – Asymptotes"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find vertical and horizontal (or oblique) asymptotes of a rational function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Vertical Asymptote:"},{"type":"text","text":" Set denominator equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (excluding common factors)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Horizontal Asymptote:"},{"type":"text","text":" Compare degrees of numerator and denominator."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor numerator and denominator if possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set denominator equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find vertical asymptotes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare degrees of numerator and denominator to determine horizontal asymptote."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If degrees are equal, horizontal asymptote is "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{\\text{leading coefficient of numerator}}{\\text{leading coefficient of denominator}}"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical asymptotes: "},{"type":"inlineMath","attrs":{"latex":"x = -8"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 4"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal asymptote: "},{"type":"inlineMath","attrs":{"latex":"y = 1"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Found by factoring and comparing degrees."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Find all the asymptotes of "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x^2 + 2x + 3}{10x^2 - 24x}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and label them appropriately."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Functions – Asymptotes"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find vertical and horizontal (or oblique) asymptotes of a rational function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Vertical Asymptote:"},{"type":"text","text":" Set denominator equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (excluding common factors)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Horizontal Asymptote:"},{"type":"text","text":" Compare degrees of numerator and denominator."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor numerator and denominator if possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set denominator equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find vertical asymptotes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare degrees of numerator and denominator to determine horizontal asymptote."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If degrees are equal, horizontal asymptote is "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{\\text{leading coefficient of numerator}}{\\text{leading coefficient of denominator}}"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical asymptotes: "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 2.4"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal asymptote: "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{1}{10}"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Found by factoring and comparing degrees."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Find the x- and y-intercepts of "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x^2 - 2x - 3}{3x^2 - 2x + 1}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and label them appropriately."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Intercepts of Rational Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find where the graph crosses the x-axis (x-intercepts) and y-axis (y-intercept)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"x-intercept:"},{"type":"text","text":" Set numerator equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (as long as denominator is not zero)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"y-intercept:"},{"type":"text","text":" Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set numerator equal to zero: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 2x - 3 = 0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check that these "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values do not make the denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For the y-intercept, substitute "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" and simplify."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":