Question

use synthetic division to simplify $\frac{x^{4}-2x^{3}-8x^{2}-5x + 20}{x - 4}$. 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 $x - 4$, so we use $c = 4$. The dividend coefficients are $1,-2,-8,-5,20$.

4	1	-2	-8	-5	20
	1

Step2: Bring down the first coefficient

Bring down the first coefficient 1.

4	1	-2	-8	-5	20
	1

Step3: Multiply and add

Multiply 4 by 1 to get 4, add to -2 gives 2.

4	1	-2	-8	-5	20
	1	2

Step4: Repeat multiplication and addition

Multiply 4 by 2 to get 8, add to -8 gives 0.

4	1	-2	-8	-5	20
	1	2	0

Step5: Repeat again

Multiply 4 by 0 to get 0, add to -5 gives -5.

4	1	-2	-8	-5	20
	1	2	0	-5

Step6: One - last multiplication and addition

Multiply 4 by -5 to get -20, add to 20 gives 0.

4	1	-2	-8	-5	20
	1	2	0	-5	0

The quotient polynomial $q(x)=x^{3}+2x^{2}-5$ and the remainder $r = 0$.

Answer:

$x^{3}+2x^{2}-5+\frac{0}{x - 4}=x^{3}+2x^{2}-5$