Is x + 2 a factor of a* + Зa? — 3х — 10?Select an answerO The remainder when you divide isIs 2x3 a factor of 32x³ – 112x4 – 72x³ – 40x² + 2x + 654?-Select an answerO The remainder when you divide isIs x6 a factor of x* + 8x° + 4x?2х + 276?Select an answerO The remainder when you divide is

Answered: 3 Is x + 2 a factor of a* + 3a* - Зх —…

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.7: Polynomial Long Division

Problem 7OE

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Question

Is x + 2 a factor of a* + Зa? — 3х — 10?
Select an answer
O The remainder when you divide is
Is 2x
3 a factor of 32x³ – 112x4 – 72x³ – 40x² + 2x + 654?
-
Select an answer
O The remainder when you divide is
Is x
6 a factor of x* + 8x° + 4x?
2х + 276?
Select an answer
O The remainder when you divide is

Expert Solution

Step 1

Check whether $x + 2$ is a factor of the polynomial by dividing $x^{3} + 3 x^{2} - 3 x - 10$ by $x + 2$ .

For that use the synthetic division method.

$\begin{matrix} - 2 & | & 1 & 3 & - 3 & - 10 \\ | & - 2 & - 2 & 10 \end{matrix} \begin{matrix} 1 & 1 & - 5 & 0 \end{matrix}$

Thus,

$\begin{array}{rcl} \frac{x^{3} + 3 x^{2} - 3 x - 10}{x + 2} & = & \frac{(x + 2) (x^{2} + x - 5)}{x + 2} \\ = & x^{2} + x - 5 \end{array}$ .

Therefore, $x + 2$ is a factor of the polynomial $x^{3} + 3 x^{2} - 3 x - 10$ .

Step 2

Check whether $2 x - 3$ is a factor of the polynomial by dividing $32 x^{5} - 112 x^{4} - 72 x^{3} - 40 x^{2} + 2 x + 654$ by $2 x - 3$ .

First divide throughout by 2 on both dividend and divisor and simplify. Then, divide the polynomials using synthetic division as follows.

$\frac{3}{2} | 16 - 56 - 36 - 20 1 327 | 24 - 48 - 126 - 219 - 327 16 - 32 - 84 - 146 - 218 0$

That is,

$\begin{array}{rcl} \frac{32 x^{5} - 112 x^{4} - 72 x^{3} - 40 x^{2} + 2 x + 654}{2 x - 3} & = & \frac{16 x^{5} - 56 x^{4} - 36 x^{3} - 20 x^{2} + x + 327}{x - \frac{3}{2}} \\ = & 16 x^{4} - 32 x^{3} - 84 x^{2} - 14673 x - 218 \end{array}$

Therefore, $2 x - 3$ is a factor of the polynomial $32 x^{5} - 112 x^{4} - 72 x^{3} - 40 x^{2} + 2 x + 654$ .

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