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Chapter 2: Problem 96

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}-4 x^{2}+1\) (a) Upper: \(x=4\) (b) Lower: \(x=-1\)

Short Answer

Expert verified
After performing the synthetic division and analysing the results, the conclusion might vary depending on the signs of the remainder coefficients. However, each result will conclude whether \(x=4\) is the upper bound and whether \(x=-1\) is the lower bound for \(f(x)=x^{3}-4 x^{2}+1\).

Step by step solution

01

Verify the Upper Bound

Start by applying synthetic division to \(f(x)\) using the specified upper bound \(x=4\). Do this by drawing a synthetic division tableau, with '4' on the outside and the coefficients of the function \(f(x)\) on the inside, which are 1, -4, and 1. Then, perform the synthetic division operation.
02

Analyze the Result for Upper Bound

Examine the results of the synthetic division process. If all the coefficients in the last row (the remainder coefficients) are non-negative, then the provided number is an upper bound for the real zeros of the function. But if any of the remainder coefficients are negative, then the provided number is not an upper bound.
03

Verify the Lower Bound

Proceed to apply synthetic division to \(f(x)\) using the specified lower bound \(x=-1\). Repeat the steps in Step 1, replacing '4' with '-1'.
04

Analyze the Result for Lower Bound

Study the results of the synthetic division process. If the remainder coefficients alternate in sign (starting from positive), then the provided number is a lower bound for the real zeros of the function. However, if the coefficients do not alternate in sign, then the provided number is not a lower bound.

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