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Chapter 3: Problem 62

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?

Short Answer

Expert verified
The Remainder Theorem can be used to find \(f(-6)\) by dividing the function \(f(x)=x^{4}+7x^{3}+8x^{2}+11x+5\) by \(x + 6\). The remainder of the division would be \(f(-6)\). It simplifies the arithmetic process, particularly when dealing with high degree functions or functions with negative roots, and hence is preferable in such situations.

Step by step solution

01

Understand the concept of the Remainder Theorem

The Remainder Theorem states that if you divide a polynomial function \(f(x)\) by the binomial \(x-c\), the remainder will be \(f(c)\). In other words, if the function \(f(x)\) is divided by \(x + 6\), the remainder is \(f(-6)\). So to compute \(f(-6)\) you just have to divide \(f(x)\) by \(x+6)\).
02

Perform the Division

Divide \(f(x)=x^{4}+7x^{3}+8x^{2}+11x+5\) by \(x+6\). By doing the division, we can see if there's any remainder. Because our goal is just to find the remainder, we perform synthetic division.
03

Evaluate the Remainder

By performing synthetic division, the remainder that we get will be our desired value of \(f(-6)\). This is, in fact, the value you would have obtained if you plugged -6 directly into the equation for \(f(x)\), but it is often easier and quicker to do the synthetic division.
04

Recognize the Advantage of the Remainder Theorem

The advantage of using the Remainder Theorem rather than evaluating \(f(-6)\) directly is mainly that it simplifies the arithmetic. Because -6 is a negative number and the function is of the 4th degree, calculations might get messy when plugging -6 in the function directly. The Remainder Theorem enables you to dodge this complex calculation, and reach the result by a more straightforward, less complicated means. Hence, it can be a helpful tool when dealing with complex polynomials functions.

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