Zeros of Polynomial Functions: Learn It 1

  • Use the Remainder Theorem to find polynomial values and the Factor Theorem to find where polynomials equal zero
  • Apply the Rational Zero Theorem to find fraction or whole number solutions, and use Descartes’ Rule of Signs to guess how many positive and negative solutions exist
  • Use the Linear Factorization Theorem to create polynomials when you know their solutions
  • Solve real-world problems using polynomial equations

Remainder Theorem

Remember how, in our synthetic division example in the previous section, we found that dividing [latex]5x^2-3x-36[/latex] by [latex]x-3[/latex] resulted in a quotient of [latex]5x+12[/latex] with a reminder of [latex]0[/latex]? This connection between division and remainders brings us to the Remainder Theorem.

Remainder Theorem

If a polynomial [latex]f(x)[/latex] is divided by [latex]x-k[/latex], then the remainder is the value [latex]f(k)[/latex].

Let’s walk through the proof of the theorem.

Recall that the Division Algorithm states that, given a polynomial dividend [latex]f(x)[/latex] and a non-zero polynomial divisor [latex]d(x)[/latex], there exist unique polynomials [latex]g(x)[/latex] and [latex]r(x)[/latex] such that

[latex]f(x) = d(x)g(x) + r(x)[/latex]

and either [latex]r(x) = 0[/latex] or the degree of [latex]r(x)[/latex] is less than the degree of [latex]d(x)[/latex]. In practice divisors, [latex]d(x)[/latex] will have degrees less than or equal to the degree of [latex]f(x)[/latex]. If the divisor, [latex]d(x)[/latex], is [latex]x - k[/latex], this takes the form

[latex]f(x) = (x - k)g(x) + r[/latex]

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

[latex]\begin{array}{rl} f(k) & = (k - k)g(k) + r \\ & = 0 \cdot g(k) + r \\ & = r \end{array}[/latex]

In other words, [latex]f(k)[/latex] is the remainder obtained by dividing [latex]f(x)[/latex] by [latex]x - k[/latex].

How to: Given a polynomial function [latex]f[/latex], evaluate [latex]f(x)[/latex] at [latex]x=k[/latex] using the Remainder Theorem.

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

This powerful theorem allows us to quickly find the remainder of polynomial division without completing the entire division process.

For example, in our case, we divided [latex]5x^2-3x-36[/latex] by [latex]x-3[/latex]. According to the Remainder Theorem, the remainder is [latex]f(3)[/latex].
[latex]\\[/latex]
By substituting [latex]3[/latex] into the polynomial [latex]f(x) = 5x^2-3x-36[/latex], we get:

[latex]f(3) = 5(3)^2 - 3(3) - 36 = 45 - 9 - 36 = 0[/latex]

This confirms that the remainder is indeed [latex]0[/latex], just as we found using synthetic division.

Use the Remainder Theorem to evaluate [latex]f\left(x\right)=6{x}^{4}-{x}^{3}-15{x}^{2}+2x - 7[/latex] at [latex]x=2[/latex].