Q. 43 Find the real zeros of f. Use th... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q. 43 (page 209)

Find the real zeros of f. Use the real zeros to factor f.
$f (x) = 3 x^{3} + 4 x^{2} + 4 x + 1$

Short Answer

Expert verified

The only real zero of the function is $- \frac{1}{3}$ .

The function can be written in factored form as $f (x) = (3 x + 1) (x^{2} + x + 1)$ .

Step by step solution

Step 1. Find the possible number of zeros

The given function $f (x) = 3 x^{3} + 4 x^{2} + 4 x + 1$ is of degree three, so it has at most three real zeros.

Step 2. Use the rational zero theorem

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

The factors of the constant term $1$ are

$p : 卤 1$

The factors of the leading coefficient $3$ are

$q : 卤 1, 卤 3$

So the possible rational zeros are

$\frac{p}{q} : 卤 1, 卤 \frac{1}{3}$

Step 3. Find the function values

For the possible rational zeros find the function value for which the function is zero and we get

$f (- 1) = 3 路 {(- 1)}^{3} + 4 路 {(- 1)}^{2} + 4 路 (- 1) + 1 f (- 1) = - 3 + 4 - 4 + 1 f (- 1) = - 3 f (- \frac{1}{3}) = 3 路 {(- \frac{1}{3})}^{3} + 4 {(- \frac{1}{3})}^{2} + 4 (- \frac{1}{3}) + 1 f (- \frac{1}{3}) = - \frac{1}{9} + \frac{4}{9} - \frac{4}{3} + 1 f (- \frac{1}{3}) = 0$

So for $x = - \frac{1}{3}$ the function value is zero, so it is a real zero and $3 x + 1$ is a factor of the function.

Step 4. Perform synthetic

Dividing the function $3 x^{3} + 4 x^{2} + 4 x + 1$ by $3 x + 1$ using synthetic division

So the quotient $x^{2} + x + 1$ is the depressed polynomial and it is a prime polynomial, so it cannot be factored further. Thus the function has only one real zero and itis written in factored form as

$f (x) = (3 x + 1) (x^{2} + x + 1)$

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91影视

Find the real zeros of f. Use the real zeros to factor f.
$f (x) = 3 x^{3} + 4 x^{2} + 4 x + 1$

Short Answer

Step by step solution

Step 1. Find the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Find the function values

Step 4. Perform synthetic

One App. One Place for Learning.

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91影视

Find the real zeros of f. Use the real zeros to factor f.f(x)=3x3+4x2+4x+1

Short Answer

Step by step solution

Step 1. Find the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Find the function values

Step 4. Perform synthetic

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the real zeros of f. Use the real zeros to factor f.
$f (x) = 3 x^{3} + 4 x^{2} + 4 x + 1$