Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

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

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Finding the possible number of zeros

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

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

$p : \pm 1, \pm 2$

The factors of the leading coefficient $4$ are

$q : \pm 1, \pm 2, \pm 4$

So, the possible rational zeros are

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

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, $f (2) = 0$

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

Use synthetic division to factor f

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

Step 5. Factor the depressed polynomial

Put depressed equation to zero and factor by grouping

$4 x^{4} - 1 = 0 4 x^{4} = 1 x^{4} = \frac{1}{4} x = \pm \frac{1}{\sqrt{2}}$

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

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

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

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

Step by Step Solution

Step 1. Finding the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Graph the polynomial function

Step 4. Finding the factor

Step 5. Factor the depressed polynomial

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

In Problems 39–56, find the real zeros of f. Use the real zeros to factor f. f(x)=4x5-8x4-x+2

Step by Step Solution

Step 1. Finding the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Graph the polynomial function

Step 4. Finding the factor

Step 5. Factor the depressed polynomial

Most popular questions for Math Textbooks

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