Question

use synthetic division to find $(x^{4}-10x^{2}+3x)div(x - 3).$ write your answer in the form $q(x)+\frac{r}{d(x)}$, where $q(x)$ is a polynomial, $r$ is an integer, and $d(x)$ is a linear polynomial. simplify any fractions.

Explanation:

Step1: Set up synthetic division

The divisor is \(x - 3\), so we use \(c=3\). The dividend is \(x^{4}-10x^{2}+3x\), which can be written in standard - form with coefficients \(1,0, - 10,3,0\).

3	1	0	- 10	3	0
	1

Step2: Bring down the first coefficient

Bring down the first coefficient \(1\).

3	1	0	- 10	3	0
	1

Step3: Multiply and add

Multiply \(3\times1 = 3\), add to the second coefficient: \(0 + 3=3\).

3	1	0	- 10	3	0
	1	3

Step4: Repeat the multiply - and - add process

Multiply \(3\times3 = 9\), add to the third coefficient: \(-10 + 9=-1\).

3	1	0	- 10	3	0
	1	3	- 1

Step5: Continue the process

Multiply \(3\times(-1)=-3\), add to the fourth coefficient: \(3+( - 3)=0\).

3	1	0	- 10	3	0
	1	3	- 1	0

Step6: Final multiply - and - add

Multiply \(3\times0 = 0\), add to the fifth coefficient: \(0+0 = 0\).

3	1	0	- 10	3	0
	1	3	- 1	0	0

The quotient \(q(x)=x^{3}+3x^{2}-x\) and the remainder \(r = 0\), and \(d(x)=x - 3\).

Answer:

\(x^{3}+3x^{2}-x+\frac{0}{x - 3}=x^{3}+3x^{2}-x\)