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Q. 55

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Found in: Page 209

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 39–56, find the real zeros of f. Use the real zeros to factor f. $f\left(x\right)=4{x}^{5}-8{x}^{4}-x+2$

The real zeros of f are $2,-\frac{1}{\sqrt{2}},and\frac{1}{\sqrt{2}}.$

The factored form of f is $\left(2{x}^{2}+1\right)\left(\sqrt{2}x-1\right)\left(\sqrt{2}x+1\right)\left(x-2\right).$

See the step by step solution

## Step 1. Finding the possible number of zeros

The given function is $f\left(x\right)=4{x}^{5}-8{x}^{4}-x+2$

The given function is of degree five, so it has at most five real zeros.

## Step 2. Use the rational zero theorem

As all the coefficients are integers so we use the rational zeros theorem.

The factors of the constant term $2$ are

$p:±1,±2$

The factors of the leading coefficient $4$ are

$q:±1,±2,±4$

So, the possible rational zeros are

$\frac{p}{q}:±1,±2,±\frac{1}{2},±\frac{1}{4}$

## Step 3. Graph the polynomial function

The graph is

From the graph, we conclude that it has three roots.

## Step 4. Finding the factor

Since $2$ appears to be zero and a potential rational zero also.

By evaluating we get, $f\left(2\right)=0$

Thus, $\left(x-2\right)$ is a factor of f.

Use synthetic division to factor f

So, factor f is localid="1646140146826" $\left(x-2\right)\left(4{x}^{4}-1\right)$

## Step 5. Factor the depressed polynomial

Put depressed equation to zero and factor by grouping

$4{x}^{4}-1=0\phantom{\rule{0ex}{0ex}}4{x}^{4}=1\phantom{\rule{0ex}{0ex}}{x}^{4}=\frac{1}{4}\phantom{\rule{0ex}{0ex}}x=±\frac{1}{\sqrt{2}}$

Therefore, the other zeros are $-\frac{1}{\sqrt{2}}and\frac{1}{\sqrt{2}}.$

The factored form of f is $\left(2{x}^{2}+1\right)\left(\sqrt{2}x-1\right)\left(\sqrt{2}x+1\right)\left(x-2\right).$