{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1a. What is the degree of the polynomial defined by: "},{"type":"inlineMath","attrs":{"latex":"p(x) = 5x(x+2)^3(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: Degree of a Polynomial"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to determine the degree of a polynomial expressed in factored form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: The highest power of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" when the polynomial is fully expanded."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factored Form: Multiply the exponents of each factor to find the total degree."}]}]}]},{"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 each factor in the polynomial: "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"(x+2)^3"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"(x-1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the degree contributed by each factor: "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" contributes 1, "},{"type":"inlineMath","attrs":{"latex":"(x+2)^3"}},{"type":"text","text":" contributes 3, and "},{"type":"inlineMath","attrs":{"latex":"(x-1)"}},{"type":"text","text":" contributes 1."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the degrees together to find the total degree."}]}]}]},{"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: 5"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The degree is "},{"type":"inlineMath","attrs":{"latex":"1 + 3 + 1 = 5"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"When you expand the polynomial, the highest power of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" will be "},{"type":"inlineMath","attrs":{"latex":"x^5"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1b. What is the end behavior model of the polynomial defined as: "},{"type":"inlineMath","attrs":{"latex":"g(x) = -2x^4 + 3x^2 - 4x + 10"}},{"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: End Behavior of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze the leading term to determine how the polynomial behaves as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches positive and negative infinity."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"End Behavior: Determined by the leading term (highest degree term)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Leading Term: "},{"type":"inlineMath","attrs":{"latex":"-2x^4"}}]}]}]},{"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 leading term: "},{"type":"inlineMath","attrs":{"latex":"-2x^4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Note the degree (even) and the sign of the leading coefficient (negative)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that for even degree polynomials, both ends go in the same direction; the sign determines whether they go up or down."}]}]}]},{"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: As "},{"type":"inlineMath","attrs":{"latex":"x \\to \\pm\\infty"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"g(x) \\to -\\infty"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Because the leading term is negative and the degree is even, both ends of the graph go down."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1c. What are the x-intercepts for the polynomial: "},{"type":"inlineMath","attrs":{"latex":"h(x) = x(x-1)^2(x+5)^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: Finding x-intercepts (roots) of a polynomial"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that make the polynomial zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"x-intercepts: Values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"h(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factored Form: Set each factor equal to 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 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x-1 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x+5 = 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":" in each equation to find the intercepts."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List the x-intercepts found from each factor."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"x = -5"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each factor gives an x-intercept. The multiplicity of each root affects the graph's behavior at those points."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1d. If a third degree polynomial has a real zero at "},{"type":"inlineMath","attrs":{"latex":"x=2"}},{"type":"text","marks":[{"type":"bold"}],"text":" and a complex zero at "},{"type":"inlineMath","attrs":{"latex":"x=3+2i"}},{"type":"text","marks":[{"type":"bold"}],"text":", where is the other 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: Fundamental Theorem of Algebra and Complex Conjugate Roots"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how complex roots appear in conjugate pairs for polynomials with real coefficients."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complex Conjugate: If "},{"type":"inlineMath","attrs":{"latex":"a+bi"}},{"type":"text","text":" is a root, "},{"type":"inlineMath","attrs":{"latex":"a-bi"}},{"type":"text","text":" is also a root."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: Number of total roots (including multiplicity)."}]}]}]},{"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":"Recall that a third degree polynomial has three roots."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Given one real root ("},{"type":"inlineMath","attrs":{"latex":"x=2"}},{"type":"text","text":") and one complex root ("},{"type":"inlineMath","attrs":{"latex":"x=3+2i"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For polynomials with real coefficients, complex roots must occur in conjugate pairs."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the conjugate of "},{"type":"inlineMath","attrs":{"latex":"3+2i"}},{"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":"inlineMath","attrs":{"latex":"x = 3 - 2i"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The other zero is the complex conjugate of "},{"type":"inlineMath","attrs":{"latex":"3+2i"}},{"type":"text","text":", which is "},{"type":"inlineMath","attrs":{"latex":"3-2i"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1e. A polynomial is known to have degree 8. What is the maximum number of real zeros the polynomial will have?"}]},{"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: Real Zeros of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between the degree of a polynomial and the number of real zeros it can have."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: The maximum number of zeros (real or complex) is equal to the degree."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Maximum Real Zeros: A polynomial of degree "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" can have up to $n$ real 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":"Recall that the degree of the polynomial is 8."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The maximum number of real zeros is equal to the degree."}]}]}]},{"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: 8"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"A degree 8 polynomial can have up to 8 real zeros."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1f. Compute and simplify the following expression: "},{"type":"inlineMath","attrs":{"latex":"(2 + 3i)(2 - 3i)"}}]},{"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: Complex Numbers and Multiplication"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to multiply complex numbers and simplify the result."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complex Number: "},{"type":"inlineMath","attrs":{"latex":"a+bi"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplication: Use the distributive property (FOIL method)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}}]}]}]},{"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":"Expand the expression using the distributive property: "},{"type":"inlineMath","attrs":{"latex":"(2 + 3i)(2 - 3i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each term: "},{"type":"inlineMath","attrs":{"latex":"2 \\times 2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"2 \\times -3i"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"3i \\times 2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"3i \\times -3i"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms and use "},{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}},{"type":"text","text":" to 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: 13"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(2 + 3i)(2 - 3i) = 4 - 9i^2 = 4 + 9 = 13"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying a complex number by its conjugate results in a real number."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1g. Simplify the following expression: "},{"type":"inlineMath","attrs":{"latex":"i^6"}}]},{"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: Powers of the Imaginary Unit "},{"type":"inlineMath","attrs":{"latex":"i"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the cyclical nature of powers of "},{"type":"inlineMath","attrs":{"latex":"i"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"i^1 = i"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"i^3 = -i"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"i^4 = 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Powers repeat every 4: "},{"type":"inlineMath","attrs":{"latex":"i^n = i^{n \\mod 4}"}}]}]}]},{"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":"Divide 6 by 4 to find the remainder: "},{"type":"inlineMath","attrs":{"latex":"6 \\div 4 = 1"}},{"type":"text","text":" remainder $2$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"So "},{"type":"inlineMath","attrs":{"latex":"i^6 = i^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall "},{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}},{"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":"inlineMath","attrs":{"latex":"-1"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"i^6 = i^2 = -1"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Powers of "},{"type":"inlineMath","attrs":{"latex":"i"}},{"type":"text","text":" repeat every 4, so "},{"type":"inlineMath","attrs":{"latex":"i^6"}},{"type":"text","text":" is equivalent to "},{"type":"inlineMath","attrs":{"latex":"i^2"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2a. What is the degree of the polynomial "},{"type":"inlineMath","attrs":{"latex":"p(x) = 2x(x+2)^3(x-1)^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: Degree of a Polynomial"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the degree of a polynomial given in factored form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: The sum of the exponents of each 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":"Identify the degree contributed by each factor: "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (1), "},{"type":"inlineMath","attrs":{"latex":"(x+2)^3"}},{"type":"text","text":" (3), "},{"type":"inlineMath","attrs":{"latex":"(x-1)^2"}},{"type":"text","text":" (2)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the degrees together to find the total degree."}]}]}]},{"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: 6"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The degree is "},{"type":"inlineMath","attrs":{"latex":"1 + 3 + 2 = 6"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2b. What is the end behavior model for the given polynomial "},{"type":"inlineMath","attrs":{"latex":"p(x) = 2x(x+2)^3(x-1)^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: End Behavior of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze the leading term and determine the end behavior of the polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"End Behavior: Determined by the leading term and degree."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Leading Coefficient: 2 (positive)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: 6 (even)"}]}]}]},{"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 degree (even) and leading coefficient (positive)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that for even degree polynomials with positive leading coefficient, both ends go up."}]}]}]},{"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: As "},{"type":"inlineMath","attrs":{"latex":"x \\to \\pm\\infty"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"p(x) \\to \\infty"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Both ends of the graph rise because the degree is even and the leading coefficient is positive."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2c. What are the zeros (roots) of the polynomial, their multiplicity, and root behavior (bounce/wiggle/cross)? 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What is the maximum number of turning points the graph of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" will have?"}]},{"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: Turning Points of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between the degree of a polynomial and its turning points."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Turning Points: Maximum number is degree minus one."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: 6"}]}]}]},{"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":"Recall the formula: Maximum turning points = degree - 1."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract 1 from the degree to find the maximum number."}]}]}]},{"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: 5"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"A degree 6 polynomial can have up to 5 turning points."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2e. Where is the y-intercept?"}]},{"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: Finding the y-intercept of a polynomial"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the y-intercept by evaluating the polynomial at "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y-intercept: "},{"type":"inlineMath","attrs":{"latex":"p(0)"}}]}]}]},{"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":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":" into the polynomial: "},{"type":"inlineMath","attrs":{"latex":"p(0) = 2 \\times 0 \\times (0+2)^3 \\times (0-1)^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate each term step by step."}]}]}]},{"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: 0"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The y-intercept is at "},{"type":"inlineMath","attrs":{"latex":"p(0) = 0"}},{"type":"text","text":" because the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" factor makes the whole expression zero."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2f. Graph a rough sketch of the polynomial on the coordinate grid provided."}]},{"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: Graphing Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to sketch the graph of a polynomial using its roots, multiplicity, and end behavior."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Roots: "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=-2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplicity: 1 (cross), 3 (wiggle), 2 (bounce)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"End Behavior: Both ends up"}]}]}]},{"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":"Mark the roots on the x-axis: "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=-2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Indicate the behavior at each root: cross at "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":", wiggle at "},{"type":"inlineMath","attrs":{"latex":"x=-2"}},{"type":"text","text":", bounce at "},{"type":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw the end behavior: both ends rising."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the graph connecting these features."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"image","attrs":{"src":"https://static.studychannel.pearsonprd.tech/study_guide_files/precalculus/sub_images/a1b2e368_image_1.png","alt":"blank coordinate grid for graphing","title":null,"width":null,"height":null}}]},{"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":"Sketch the graph with roots at "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=-2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","text":", showing the correct behavior at each root and both ends rising."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the provided grid to draw your rough sketch."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Form a polynomial in standard form that has degree 3 and zeros at "},{"type":"inlineMath","attrs":{"latex":"x=2"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","marks":[{"type":"bold"}],"text":", and "},{"type":"inlineMath","attrs":{"latex":"x=-1"}},{"type":"text","marks":[{"type":"bold"}],"text":". Let the leading coefficient be 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 Roots"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to write a polynomial given its roots and leading coefficient."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard Form: "},{"type":"inlineMath","attrs":{"latex":"ax^3 + bx^2 + cx + d"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Roots: "},{"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":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factored Form: "},{"type":"inlineMath","attrs":{"latex":"(x-2)(x-1)(x+1)"}}]}]}]},{"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 polynomial in factored form: "},{"type":"inlineMath","attrs":{"latex":"(x-2)(x-1)(x+1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the product to get the standard form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Ensure the leading coefficient 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":"inlineMath","attrs":{"latex":"x^3 - 2x^2 - x + 2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expanding "},{"type":"inlineMath","attrs":{"latex":"(x-2)(x-1)(x+1)"}},{"type":"text","text":" gives the standard form with leading coefficient 1."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Use any method you know to determine if "},{"type":"inlineMath","attrs":{"latex":"x-2"}},{"type":"text","marks":[{"type":"bold"}],"text":" is a factor of the polynomial "},{"type":"inlineMath","attrs":{"latex":"p(x) = x^3 - 2x^2 + 3x - 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: Factor Theorem and Remainder Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the Factor Theorem or Remainder Theorem to check if a given binomial is a factor."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor Theorem: "},{"type":"inlineMath","attrs":{"latex":"x-a"}},{"type":"text","text":" is a factor if "},{"type":"inlineMath","attrs":{"latex":"p(a) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remainder Theorem: The remainder when dividing by "},{"type":"inlineMath","attrs":{"latex":"x-a"}},{"type":"text","text":" is "},{"type":"inlineMath","attrs":{"latex":"p(a)"}},{"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 "},{"type":"inlineMath","attrs":{"latex":"x-2=0"}},{"type":"text","text":" so "},{"type":"inlineMath","attrs":{"latex":"x=2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":"p(2)"}},{"type":"text","text":" by substituting "},{"type":"inlineMath","attrs":{"latex":"x=2"}},{"type":"text","text":" into the polynomial."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If "},{"type":"inlineMath","attrs":{"latex":"p(2)=0"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"x-2"}},{"type":"text","text":" is a factor."}]}]}]},{"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: Yes, "},{"type":"inlineMath","attrs":{"latex":"x-2"}},{"type":"text","marks":[{"type":"bold"}],"text":" is a factor"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"p(2) = 2^3 - 2 \\times 2^2 + 3 \\times 2 - 6 = 0"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"x-2"}},{"type":"text","text":" is a factor."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5a. What is the domain of the polynomial "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2x^3 + 5x^2 - 3x + 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: Domain of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the domain of polynomial functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: All real numbers for polynomials."}]}]}]},{"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":"Recall that polynomials are defined for all real values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"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":"inlineMath","attrs":{"latex":"(-\\infty, \\infty)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The domain of any polynomial is all real numbers."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5b. What is the range of the polynomial "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2x^3 + 5x^2 - 3x + 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: Range of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the range of cubic polynomials."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: For odd degree polynomials, the range is all real numbers."}]}]}]},{"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":"Note that the polynomial is cubic (degree 3, odd)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that odd degree polynomials have range "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, \\infty)"}},{"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":"inlineMath","attrs":{"latex":"(-\\infty, \\infty)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The range of a cubic polynomial is all real numbers."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5c. What is the maximum number of real zeros for this polynomial?"}]},{"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: Real Zeros of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between degree and real zeros."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: Maximum number of real zeros is equal to the degree."}]}]}]},{"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 degree of the polynomial (3)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The maximum number of real zeros is 3."}]}]}]},{"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: 3"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"A cubic polynomial can have up to 3 real zeros."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5d. Where is the y-intercept for this polynomial?"}]},{"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: Finding the y-intercept"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the y-intercept by evaluating "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y-intercept: "},{"type":"inlineMath","attrs":{"latex":"f(0)"}}]}]}]},{"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":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":" into the polynomial: "},{"type":"inlineMath","attrs":{"latex":"f(0) = -2 \\times 0^3 + 5 \\times 0^2 - 3 \\times 0 + 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the expression."}]}]}]},{"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: 1"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The y-intercept is at "},{"type":"inlineMath","attrs":{"latex":"f(0) = 1"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6a. Find all possible rational zeros of the polynomial "},{"type":"inlineMath","attrs":{"latex":"p(x) = x^3 + 4x^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: Rational Root Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the Rational Root Theorem to list possible rational zeros."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational Root Theorem: Possible 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":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Constant term: "},{"type":"inlineMath","attrs":{"latex":"-6"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Leading coefficient: $1$"}]}]}]},{"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":"List all factors of the constant term "},{"type":"inlineMath","attrs":{"latex":"-6"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List all factors of the leading coefficient $1$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Form all possible "},{"type":"inlineMath","attrs":{"latex":"\\pm \\frac{p}{q}"}},{"type":"text","text":" values."}]}]}]},{"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"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\pm2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\pm3"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\pm6"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are all the possible rational zeros according to the Rational Root Theorem."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6b. Use the Remainder Theorem and/or Factor Theorem to test your possible rational zeros and identify one of the Real Zeros of the polynomial."}]},{"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: Testing Rational Zeros"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to substitute possible zeros and find which one is a real zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remainder Theorem: Substitute each possible zero into "},{"type":"inlineMath","attrs":{"latex":"p(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If "},{"type":"inlineMath","attrs":{"latex":"p(a) = 0"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"x-a"}},{"type":"text","text":" is a 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":"Substitute each possible zero into "},{"type":"inlineMath","attrs":{"latex":"p(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check which value makes "},{"type":"inlineMath","attrs":{"latex":"p(x) = 0"}},{"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":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","marks":[{"type":"bold"}],"text":" is a real zero"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substituting "},{"type":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"p(x)"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"p(1) = 1 + 4 + 1 - 6 = 0"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6c. Rewrite your polynomial "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" in factored form using the information you found above about the Real Zeros and polynomial or synthetic division."}]},{"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 Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to factor a cubic polynomial given one real zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factored Form: "},{"type":"inlineMath","attrs":{"latex":"g(x) = (x - a)(quadratic)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use synthetic or polynomial division to find the quadratic 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":"Divide "},{"type":"inlineMath","attrs":{"latex":"p(x)"}},{"type":"text","text":" by "},{"type":"inlineMath","attrs":{"latex":"x-1"}},{"type":"text","text":" using synthetic or polynomial division."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the result as "},{"type":"inlineMath","attrs":{"latex":"(x-1)(quadratic)"}},{"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":"inlineMath","attrs":{"latex":"(x-1)(x^2 + 5x + 6)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factoring further: "},{"type":"inlineMath","attrs":{"latex":"(x-1)(x+2)(x+3)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The polynomial is fully factored."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6d. Graph the resulting Polynomial on the coordinate grid provided."}]},{"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: Graphing Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to sketch the graph of a cubic polynomial using its roots and end behavior."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Roots: "},{"type":"inlineMath","attrs":{"latex":"x=1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=-2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x=-3"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"End Behavior: Cubic with positive leading coefficient"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type"