Q1. Solve the inequality: 36(x^2 −1) > 65x

Background

Topic: Solving Polynomial Inequalities

This question tests your ability to solve quadratic inequalities by rearranging terms, factoring, and analyzing sign changes.

Key Terms and Formulas

• Quadratic Inequality: An inequality involving a quadratic expression.

• Factoring: Writing a polynomial as a product of its factors.

• Zero Product Property: If , then or .

Step-by-Step Guidance

1. Move all terms to one side to set the inequality to zero: .

2. Expand and simplify the expression: .

3. Rearrange the terms in standard quadratic form: .

4. Factor the quadratic expression, if possible, or use the quadratic formula to find the critical points (where the expression equals zero).

5. Set up a sign chart using the critical points to determine where the expression is positive.

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Q2. Find all zeros of the function and write the polynomial as a product of linear factors:

Background

Topic: Finding Zeros of Polynomials and Factoring

This question tests your ability to find all real and complex zeros of a cubic polynomial and express the polynomial as a product of linear factors.

Key Terms and Formulas

• Zero of a function: A value such that .

• Factoring: Expressing a polynomial as a product of linear (or irreducible) factors.

• Rational Root Theorem: Possible rational zeros are (factors of constant)/(factors of leading coefficient).

Step-by-Step Guidance

1. List all possible rational zeros using the Rational Root Theorem.

2. Test possible zeros by substituting them into or using synthetic division.

3. Once a zero is found, factor it out and reduce the polynomial's degree.

4. Repeat the process for the reduced polynomial to find all zeros.

5. Write the polynomial as a product of linear factors using the zeros found.

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Q3. Find such that has the factor .

Background

Topic: Factor Theorem and Polynomial Division

This question tests your understanding of the Factor Theorem, which relates zeros and factors of polynomials.

Key Terms and Formulas

• Factor Theorem: is a factor of if and only if .

Step-by-Step Guidance

1. According to the Factor Theorem, set because is a factor.

2. Substitute into : .

3. Simplify the equation to solve for .

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Q4. Form a polynomial whose zeros and degree are given. Use a leading coefficient of 1: Zeros: , , $4$; degree 3

Background

Topic: Constructing Polynomials from Zeros

This question tests your ability to write a polynomial given its zeros and degree.

Key Terms and Formulas

• If is a zero, then is a factor.

• Degree: The highest power of in the polynomial.

Step-by-Step Guidance

1. Write the factors corresponding to each zero: , , .

2. Multiply the factors together to form the polynomial.

3. Ensure the polynomial is of degree 3 and has a leading coefficient of 1.

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Q5. Form a polynomial with real coefficients having the given degree and zeros: Degree: 3; zeros: ,

Background

Topic: Constructing Polynomials with Complex Zeros

This question tests your understanding that complex zeros with real coefficients must occur in conjugate pairs.

Key Terms and Formulas

• Complex Conjugate Root Theorem: If is a zero, so is (for polynomials with real coefficients).

• Form the polynomial as the product of linear factors for each zero.

Step-by-Step Guidance

1. List all zeros: , , and (the conjugate).

2. Write the corresponding factors: , , .

3. Multiply the complex conjugate factors to get a quadratic with real coefficients.

4. Multiply the quadratic by the remaining linear factor to get the cubic polynomial.

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Q6. Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor over the real numbers.

Background

Topic: Rational Zeros Theorem and Factoring Polynomials

This question tests your ability to list possible rational zeros, test them, and factor the polynomial completely over the real numbers.

Key Terms and Formulas

• Rational Zeros Theorem: Possible rational zeros are (factors of constant)/(factors of leading coefficient).

• Factoring: Expressing the polynomial as a product of linear and/or irreducible quadratic factors.

Step-by-Step Guidance

1. List all possible rational zeros using the Rational Zeros Theorem.

2. Test each possible zero by substitution or synthetic division to find a real zero.

3. Once a real zero is found, factor it out and reduce the degree of the polynomial.

4. Repeat the process for the reduced polynomial to find all real zeros.

5. Write the factored form using the real zeros found.

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Q7. Determine the real zeros of the polynomial and their multiplicities. Then decide whether the graph touches or crosses the x-axis at each zero.

Background

Topic: Zeros, Multiplicities, and Graph Behavior

This question tests your understanding of how the multiplicity of a zero affects the graph's behavior at that zero.

Key Terms and Formulas

• Multiplicity: The number of times a zero is repeated.

• If the multiplicity is odd, the graph crosses the x-axis; if even, it touches and turns around.

Step-by-Step Guidance

1. Identify the zeros by setting each factor equal to zero: and .

2. Determine the multiplicity of each zero from the exponents in the factors.

3. Decide whether the graph touches or crosses the x-axis at each zero based on multiplicity.

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Q8. List the potential rational zeros of the polynomial function . Do not find the zeros.

Background

Topic: Rational Zeros Theorem

This question tests your ability to list all possible rational zeros using the Rational Zeros Theorem.

Key Terms and Formulas

• Rational Zeros Theorem: Possible rational zeros are (factors of constant)/(factors of leading coefficient).

Step-by-Step Guidance

1. List all factors of the constant term (6): .

2. List all factors of the leading coefficient (-4): .

3. Form all possible fractions (in lowest terms) of (factors of 6)/(factors of -4), both positive and negative.

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