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College Algebra: Quadratic and Polynomial Functions Review Guidance

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Q1. Given the quadratic function in vertex form, answer the following for :

Background

Topic: Quadratic Functions (Vertex Form)

This question tests your understanding of the vertex form of a quadratic function, how to identify the vertex, axis of symmetry, direction of opening, intercepts, and how to describe the domain and range.

Key Terms and Formulas:

  • Vertex form:

  • Vertex:

  • Axis of symmetry:

  • Direction: If , opens up; if , opens down

  • Domain: All real numbers for quadratics

  • Range: Depends on and direction

Step-by-Step Guidance

  1. Identify , , and from the given function. Here, , , .

  2. State the vertex using .

  3. Determine if the parabola opens up or down by checking the sign of .

  4. Write the equation of the axis of symmetry ().

  5. To find -intercepts, set and solve for .

  6. To find the -intercept, set and solve for .

  7. Describe the domain and range based on the vertex and direction.

Try solving on your own before revealing the answer!

Q2. Rewrite in vertex form by completing the square, then answer parts a–g.

Background

Topic: Completing the Square & Quadratic Analysis

This question checks your ability to rewrite a quadratic in vertex form, identify key features, and analyze the graph.

Key Terms and Formulas:

  • Standard form:

  • Vertex form:

  • Completing the square: Rearranging to express as

Step-by-Step Guidance

  1. Factor from the and terms: .

  2. Complete the square inside the brackets: Add and subtract inside.

  3. Rewrite as and simplify.

  4. Identify the vertex from the vertex form.

  5. Determine the direction of opening from the sign of .

  6. Find the axis of symmetry (), -intercepts (set ), and -intercept (set ).

Try solving on your own before revealing the answer!

Q3. For , does the function have a maximum or minimum value? What is that value?

Background

Topic: Quadratic Function Maximum/Minimum

This question tests your ability to determine whether a quadratic has a maximum or minimum and to find that value.

Key Terms and Formulas:

  • If , the parabola opens up (minimum); if , opens down (maximum).

  • Vertex -coordinate:

  • Minimum/maximum value:

Step-by-Step Guidance

  1. Identify and from the quadratic: , .

  2. Determine if the parabola opens up or down by checking the sign of .

  3. Find the -coordinate of the vertex using .

  4. Plug this value into to find the minimum or maximum value.

Try solving on your own before revealing the answer!

Q4. Use the quadratic function to find the - and -intercepts.

Background

Topic: Quadratic Intercepts

This question checks your ability to find the - and -intercepts of a quadratic function.

Key Terms and Formulas:

  • -intercepts: Set and solve for .

  • -intercept: Set and solve for .

Step-by-Step Guidance

  1. Set : and solve for (use quadratic formula if needed).

  2. For the -intercept, substitute into and simplify.

Try solving on your own before revealing the answer!

Q5. A baseball's height is given by . How long does it take to reach maximum height, and what is that height?

Background

Topic: Quadratic Applications (Projectile Motion)

This question applies quadratic functions to real-world motion, specifically finding the vertex (maximum point) of a parabola.

Key Terms and Formulas:

  • Vertex -coordinate:

  • Maximum height: at this value

Step-by-Step Guidance

  1. Identify , in the quadratic.

  2. Calculate to find the time at maximum height.

  3. Substitute this value into to find the maximum height.

Try solving on your own before revealing the answer!

Q6. Use the end behavior of a polynomial graph to determine if the degree is even or odd, and if the leading coefficient is positive or negative.

Background

Topic: Polynomial End Behavior

This question tests your understanding of how the degree and leading coefficient affect the end behavior of polynomial graphs.

Key Terms and Formulas:

  • Even degree: Both ends go the same direction.

  • Odd degree: Ends go opposite directions.

  • Positive leading coefficient: Right end up; negative: right end down.

Step-by-Step Guidance

  1. Observe the graph's left and right end behavior.

  2. Decide if the ends go the same way (even) or opposite (odd).

  3. Check if the right end goes up (positive) or down (negative) to determine the sign of the leading coefficient.

Try solving on your own before revealing the answer!

Q7. Find the intercepts of .

Background

Topic: Polynomial Intercepts

This question checks your ability to find - and -intercepts for a cubic polynomial.

Key Terms and Formulas:

  • -intercept:

  • -intercepts: Set and solve for (factoring or rational root theorem may help)

Step-by-Step Guidance

  1. Find the -intercept by evaluating .

  2. Set and attempt to factor or use rational root theorem to find -intercepts.

Try solving on your own before revealing the answer!

Q8. Sketch : end behavior, -intercept, zeros and multiplicities, test point, and graph selection.

Background

Topic: Graphing Polynomial Functions

This question tests your ability to analyze and sketch a higher-degree polynomial, including end behavior, intercepts, and multiplicities.

Key Terms and Formulas:

  • End behavior: Determined by degree and leading coefficient

  • Zeros: Set each factor to zero

  • Multiplicity: Exponent of each factor

  • Test point: Plug in a value between zeros to determine sign

Step-by-Step Guidance

  1. Expand or analyze the leading term to determine end behavior.

  2. Find the -intercept by evaluating .

  3. Set each factor to zero to find real zeros and their multiplicities.

  4. Choose a test point between zeros and evaluate to determine the sign of .

  5. Use this information to match the correct graph.

Try solving on your own before revealing the answer!

Q9. Select a possible function for a given graph from a list of options.

Background

Topic: Matching Polynomial Equations to Graphs

This question tests your ability to interpret polynomial graphs and match them to possible equations based on zeros and end behavior.

Key Terms and Formulas:

  • Zeros: Where the graph crosses the -axis

  • Leading coefficient: Affects end behavior

Step-by-Step Guidance

  1. Identify the -intercepts from the graph.

  2. Check the end behavior (up/down) on both sides.

  3. Match these features to the options provided.

Try solving on your own before revealing the answer!

Q10. Use synthetic division to divide by and write in the form .

Background

Topic: Synthetic Division and Remainder Theorem

This question tests your ability to perform synthetic division and express a polynomial in division algorithm form.

Key Terms and Formulas:

  • Synthetic division: Shortcut for dividing by

  • Division algorithm:

Step-by-Step Guidance

  1. Set up synthetic division using .

  2. Write the coefficients: .

  3. Carry out synthetic division to find the quotient and remainder.

  4. Express in the required form using the results.

Try solving on your own before revealing the answer!

Q11. Use synthetic division and the remainder theorem to find the remainder when is divided by .

Background

Topic: Synthetic Division and Remainder Theorem

This question checks your ability to use synthetic division to find the remainder when dividing by .

Key Terms and Formulas:

  • Remainder theorem: is the remainder when divided by

  • Synthetic division: Use

Step-by-Step Guidance

  1. List all coefficients, including zeros for missing degrees.

  2. Set up synthetic division with .

  3. Perform the synthetic division steps to find the remainder.

Try solving on your own before revealing the answer!

Q12. Use synthetic division and the factor theorem to determine if is a factor of .

Background

Topic: Factor Theorem and Synthetic Division

This question tests your ability to use synthetic division to check if a binomial is a factor of a polynomial.

Key Terms and Formulas:

  • Factor theorem: is a factor if

  • Synthetic division: Use

Step-by-Step Guidance

  1. Set up synthetic division with and the coefficients .

  2. Perform synthetic division to find the remainder.

  3. If the remainder is zero, is a factor; otherwise, it is not.

Try solving on your own before revealing the answer!

Q13. Given and is a zero of multiplicity 2, find all remaining zeros and write in factored form.

Background

Topic: Factoring Polynomials and Multiplicity

This question checks your ability to factor a cubic polynomial given a zero and its multiplicity, and to find all zeros.

Key Terms and Formulas:

  • Multiplicity: The number of times a zero occurs

  • Factored form:

Step-by-Step Guidance

  1. Since is a zero of multiplicity 2, factor out from .

  2. Divide by to find the remaining factor.

  3. Set the remaining factor equal to zero to find the last zero.

Try solving on your own before revealing the answer!

Q14. Use the rational zeros theorem to list all possible rational zeros of .

Background

Topic: Rational Zeros Theorem

This question tests your ability to list all possible rational zeros of a polynomial using the rational zeros theorem.

Key Terms and Formulas:

  • Possible rational zeros: , where divides the constant term, divides the leading coefficient

Step-by-Step Guidance

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

  2. List all factors of the leading coefficient ($11$).

  3. Form all possible fractions using these factors.

Try solving on your own before revealing the answer!

Q15. Solve in the complex numbers.

Background

Topic: Solving Polynomial Equations (Complex Roots)

This question checks your ability to solve a cubic equation and express all roots, including complex ones.

Key Terms and Formulas:

  • Factor as a difference of cubes:

  • Quadratic formula for complex roots

Step-by-Step Guidance

  1. Recognize as a difference of cubes and factor accordingly.

  2. Solve for the real root.

  3. Solve for the complex roots using the quadratic formula.

Try solving on your own before revealing the answer!

Q16. Form a third-degree polynomial with real coefficients and leading coefficient 1, given zeros and .

Background

Topic: Constructing Polynomials from Zeros (Complex Conjugate Root Theorem)

This question tests your ability to write a polynomial given complex and real zeros, using the fact that complex zeros come in conjugate pairs for polynomials with real coefficients.

Key Terms and Formulas:

  • If is a zero, so is

  • Form:

Step-by-Step Guidance

  1. List all zeros: , , .

  2. Write the factors: , , .

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

  4. Multiply this quadratic by to get the cubic polynomial.

Try solving on your own before revealing the answer!

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