Evaluate a polynomial using the Remainder Theorem.Use the Factor Theorem to solve a polynomial equation.Use the Rational Zero Theorem to find rational zeros.Find zeros of a polynomial function.Use the Linear Factorization Theorem to find polynomials with given zeros.Use Descartes’ Rule of Signs.Solve real-world applications of polynomial equations


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A new bakery offers decorated sheet cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?

This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.


Evaluating a Polynomial Using the Remainder Theorem

In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by

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Analysis

We can check our answer by evaluating

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is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:


We can factor the quadratic factor to write the polynomial as


By the Factor Theorem, the zeros of

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has integer coefficients, then every rational zero of
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has the form
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whereis a factor of the constant term
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andis a factor of the leading coefficient
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When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.




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Given a polynomial function

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use the Rational Zero Theorem to find rational zeros.

Determine all factors of the constant term and all factors of the leading coefficient.Determine all possible values of
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whereis a factor of the constant term andis a factor of the leading coefficient. Be sure to include both positive and negative candidates.Determine which possible zeros are actual zeros by evaluating each case of
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