list all possible roots , find all the factors , state the zero f(x) = x³ + 6x² – 13x – 6

Answered: f(x) = x³ + 6x² – 13x – 6

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter4: Polynomial And Rational Functions

Section4.1: Quadratic Functions

Problem 3SC

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list all possible roots , find all the factors , state the zero

Expert Solution

Step 1

Given that $f (x) = x^{3} + 6 x^{2} - 13 x - 6$ .

To determine all the roots and factors of $f (x)$ .

For, consider $x^{3} + 6 x^{2} - 13 x - 6 = 0$ .

By trail and error method, at $x = 2$ ,

$\begin{array}{rcl} f (2) & = & 2^{3} + (6 \times 2^{2}) - (13 \times 2) - 6 \\ = & 8 + 24 - 26 - 6 \\ = & 0 \end{array}$

Therefore, 2 is a root of $f (x)$ .

Which implies that, $x - 2$ is a factor of $f (x)$ .

Now to find the other factors use long division method.

That is $\frac{x^{3} + 6 x^{2} - 13 x - 6}{x - 2}$

Therefore, $f (x) = (x - 2) (x^{2} + 8 x + 3)$

Step 2

To find other roots, consider $(x^{2} + 8 x + 3)$

By definition,

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Therefore,

$\begin{array}{rcl} x & = & \frac{- 8 \pm \sqrt{8^{2} - 4 \times 3}}{2} \\ = & \frac{- 8 \pm \sqrt{64 - 12}}{2} \\ = & \frac{- 8 \pm \sqrt{52}}{2} \\ = & - 4 \pm \sqrt{13} \end{array}$

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