Three Techniques for Evaluating and Finding Zeros of Polynomial Functions

In this section, we will apply polynomial division techniques to analyze and evaluate polynomials. You will be able to evaluate a polynomial function for a given value using the remainder theorem and the factor theorem. These two techniques work well when the roots of a polynomial are integers. We need to use the rational zeros theorem when we have rational roots. This technique also uses polynomial division but will yield zeros that are rational numbers.

Learning Objectives

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.

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-5-zeros-of-polynomial-functions
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