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In this section, you will:
  • 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

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 x k , the remainder may be found quickly by evaluating the polynomial function at k , that is, f ( k ) Let’s walk through the proof of the theorem.

Recall that the Division Algorithm    states that, given a polynomial dividend f ( x ) and a non-zero polynomial divisor d ( x ) where the degree of d ( x ) is less than or equal to the degree of f ( x ) , there exist unique polynomials q ( x ) and r ( x ) such that

f ( x ) = d ( x ) q ( x ) + r ( x )

If the divisor, d ( x ) , is x k , this takes the form

f ( x ) = ( x k ) q ( x ) + r

Since the divisor x k is linear, the remainder will be a constant, r . And, if we evaluate this for x = k , we have

f ( k ) = ( k k ) q ( k ) + r = 0 q ( k ) + r = r

In other words, f ( k ) is the remainder obtained by dividing f ( x ) by x k .

The remainder theorem

If a polynomial f ( x ) is divided by x k , then the remainder is the value f ( k ) .

Given a polynomial function f , evaluate f ( x ) at x = k using the Remainder Theorem.

  1. Use synthetic division to divide the polynomial by x k .
  2. The remainder is the value f ( k ) .

Using the remainder theorem to evaluate a polynomial

Use the Remainder Theorem to evaluate f ( x ) = 6 x 4 x 3 15 x 2 + 2 x 7 at x = 2.

To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by x 2.

The remainder is 25. Therefore, f ( 2 ) = 25.

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Use the Remainder Theorem to evaluate f ( x ) = 2 x 5 3 x 4 9 x 3 + 8 x 2 + 2 at x = 3.

f ( 3 ) = 412

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Using the factor theorem to solve a polynomial equation

The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.

f ( x ) = ( x k ) q ( x ) + r

If k is a zero, then the remainder r is f ( k ) = 0 and f ( x ) = ( x k ) q ( x ) + 0 or f ( x ) = ( x k ) q ( x ) .

Notice, written in this form, x k is a factor of f ( x ) . We can conclude if k is a zero of f ( x ) , then x k is a factor of f ( x ) .

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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