Problem 52 - College Algebra

Question

Explain how the Remainder Theorem can be used to find \(f(-6)\) if \(f(x)=x^{4}+7 x^{3}+8 x^{2}+11 x+5 .\) What advantage is there to using the Remainder Theorem in this situation rather than evaluating \(f(-6)\) directly?

Step-by-Step Solution

Answer

Using the Remainder Theorem, the value of \(f(-6)\) is found to be -79. The Remainder Theorem simplifies the calculation by avoiding the need to compute large, negative numbers that arise when evaluating \(f(-6)\) directly.

Step 1: Write down the polynomial and divisor

We have the polynomial \(f(x)=x^{4}+7x^{3}+8x^{2}+11x+5\). To get \(f(-6)\) we will find the remainder when f(x) is divided by \(x - (-6)\) or \(x + 6\).

Step 2: Perform polynomial division

The division of \(x^{4}+7x^{3}+8x^{2}+11x+5\) by \(x+6\) using synthetic division or long division yields a quotient of \(x^{3}+x^{2}-2x-14\) and a remainder of -79.

Step 3: Identify the remainder as the function value

According to the Remainder theorem, the remainder from the polynomial division is the value of the function evaluated at the divisor value. Therefore, \(f(-6)\) is equal to -79.