118 Evaluate 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
If the divisor, d(x), is x – k, this takes the form
Since the divisor x – k is linear, the remainder will be a constant, r. And, if we evaluate this for x = k, we have
In other words, f(k) is the remainder obtained by dividing f(x) by x – k.
A General Note: The Remainder Theorem
If a polynomial [latex]f\left(x\right)\\[/latex] is divided by x – k, then the remainder is the value [latex]f\left(k\right)\\[/latex].
How To: Given a polynomial function [latex]f\\[/latex], evaluate [latex]f\left(x\right)\\[/latex] at [latex]x=k\\[/latex] using the Remainder Theorem.
- Use synthetic division to divide the polynomial by [latex]x-k\\[/latex].
- The remainder is the value [latex]f\left(k\right)\\[/latex].
Example 1: Using the Remainder Theorem to Evaluate a Polynomial
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].
Solution
To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x - 2\\[/latex].
The remainder is 25. Therefore, [latex]f\left(2\right)=25\\[/latex].
Try It 1
Use the Remainder Theorem to evaluate [latex]f\left(x\right)=2{x}^{5}-3{x}^{4}-9{x}^{3}+8{x}^{2}+2\\[/latex]
at [latex]x=-3\\[/latex].
Analysis of the Solution
We can check our answer by evaluating [latex]f\left(2\right)\\[/latex].