Chapter 4: Q 39. (page 216)

In Problems 31– 40, find the complex zeros of each polynomial function. Write f in factored form.
$f (x) = 3 x^{4} - x^{3} - 9 x^{2} + 159 x - 52$

Short Answer

Expert verified

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

Complex zeros are $2 \pm 3 i .$

Step by step solution

Step 1. Given Information

The given polynomial is $f (x) = 3 x^{4} - x^{3} - 9 x^{2} + 159 x - 52$

Since $a_{0} = - 52$ and $a_{n} = 3$ .

The factors of $- 52 a r e \pm 1, \pm 2, \pm 3, \pm 4, \pm 13, \pm 26, \pm 52$ and factors of $3$ are $\pm 1, \pm 3$

Therefore, potential roots of $f (x)$ are $\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 13, \pm \frac{13}{3}, \pm 26, \pm \frac{26}{3}, \pm 52, \pm \frac{52}{3}$

Step 2. Check for zeros

First, we will use synthetic division to determine the zero of $f (x),$ let us check at $\frac{1}{3}$ . We see that

. $f (\frac{1}{3}) = 0$ That is $\frac{1}{3}$ is a zero of polynomial.

$f (x) = (x - \frac{1}{3}) (3 x^{3} - 9 x + 156)$

Step 3. Factor 3x3-9x+156

$3 x^{3} - 9 x + 156 = 0 3 (x + 4) (x^{2} - 4 x + 13) = 0$

where $x^{2} - 4 x + 13$ is not factorable.

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

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

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In Problems 31– 40, find the complex zeros of each polynomial function. Write f in factored form.
$f (x) = 3 x^{4} - x^{3} - 9 x^{2} + 159 x - 52$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Check for zeros

Step 3. Factor 3x3-9x+156

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

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In Problems 31– 40, find the complex zeros of each polynomial function. Write f in factored form.f(x)=3x4-x3-9x2+159x-52

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Check for zeros

Step 3. Factor 3x3-9x+156

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Discrete Mathematics

Decision Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

In Problems 31– 40, find the complex zeros of each polynomial function. Write f in factored form.
$f (x) = 3 x^{4} - x^{3} - 9 x^{2} + 159 x - 52$