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Chapter 2: 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 value of \(f(-6)\) can be found using the Remainder Theorem via synthetic division which simplifies the calculations. The value of \(f(-6)\) will be the remainder of this division.

Step by step solution

01

Set up the division

Arrange the coefficients of the function in descending power order. For function \(f(x)=x^{4}+7x^{3}+8x^{2}+11x+5\), this would be [1, 7, 8, 11, 5]. Now, write the value to substitute (our \(x\)) on the side. That's -6 in our case.
02

Perform Synthetic Division

Drop down the first coefficient (1), then multiply -6 by that number and write it underneath the second coefficient (7). Add the numbers in this new column together, and then repeat this process for each coefficient.
03

Interpret the result

The numbers in the last row of the synthetic division are the coefficients of the quotient, save for the final number which is the remainder. The remainder of this operation is the value of \(f(-6)\).

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