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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q 40. (page 216)

In Problems 31鈥� 40, find the complex zeros of each polynomial function. Write f in factored form.
$f (x) = 2 x^{4} + x^{3} - 35 x^{2} - 113 x + 65$

Short Answer

Expert verified

$f (x) = (x - 5) (2 x - 1) (x + 3 - 2 i) (x + 3 + 2 i)$

Complex zeros are $- 3 卤 2 i .$

Step by step solution

01

Step 1. Given Information

The given polynomial is

$f (x) = 2 x^{4} + x^{3} - 35 x^{2} - 113 x + 65$

02

Step 2. Finding potential roots

The factors of constant terms are $卤 1, 卤 5, 卤 13, 卤 65$ and factor of leading term are $卤 1, 卤 2$ .

The potential roots of $f (x)$ are $卤 1, 卤 \frac{1}{2}, 卤 5, 卤 \frac{5}{2}, 卤 13, 卤 \frac{13}{2}, 卤 15, 卤 \frac{15}{2}$

03

Step 3. Finding factors by checking zeros

Let us check at $\frac{1}{2},$ $f (\frac{1}{2}) = 0$

So, $f (x) = (x - \frac{1}{2}) (2 x^{3} + 2 x^{2} - 34 x - 130)$

Now, factor

$2 x^{3} + 2 x^{2} - 34 x - 130 = 0 2 (x - 5) (x^{2} + 6 x + 13) = 0$

Where $x^{2} + 6 x + 13$ can not factorable.

04

Step 4. Factor x2+6x+13 by quadratic formula

$x = \frac{- 6 卤 \sqrt{36 - 52}}{2} = \frac{- 6 卤 \sqrt{- 16}}{2} x = \frac{- 6 卤 4 i}{2} x = - 3 + 2 i, - 3 - 2 i$

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Most popular questions from this chapter

Solve the inequality algebraically

$(x - 1) (x - 2) (x - 3) 鈮� 0$

Use the Factor Theorem to prove that $x + c$ is a factor of

role="math" localid="1646067091231" $x^{n} + c^{n}$ if $n 鈮� 1$ is an odd integer

In Problems $7 - 44$ , follow Steps $1$ through 7 on page $228$ to analyze the graph of each function.

$G (x) = \frac{x^{3} + 1}{x^{2} + 2 x}$

In Problems 57鈥� 62, find the real zeros of f. If necessary, round to two decimal places.

$f (x) = x^{4} - 1.4 x^{3} - 33.71 x^{2} + 23.94 x + 292.41$

Find the real zeros of f. Use the real zeros to factor f.

$f (x) = 3 x^{3} + 4 x^{2} + 4 x + 1$

See all solutions

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