Question

use synthetic division to find the quotient and remainder when (-2x^{4}+10x^{3}+11x^{2}+6x - 3) is divided by (x - 6) by completing the parts below.
(a) complete this synthetic division table.
(\begin{array}{r|rrrrr}6)&-2&10&11&6&-3\\&&&&&\hline&&&&&end{array})
(b) write your answer in the following form: quotient (+\frac{\text{remainder}}{x - 6}).
(\frac{-2x^{4}+10x^{3}+11x^{2}+6x - 3}{x - 6}=square+\frac{square}{x - 6})

Explanation:

Step1: Set up synthetic division

The divisor is $x-6$, so we use $6$ for synthetic division. The coefficients of the dividend $-2x^4 + 10x^3 + 11x^2 + 6x - 3$ are $-2, 10, 11, 6, -3$.
Bring down the first coefficient:
$\boldsymbol{-2}$

Step2: Multiply and add (1st term)

Multiply the brought-down value by 6, add to next coefficient.
$(-2) \times 6 = -12$; $10 + (-12) = -2$
$\boldsymbol{-2}$

Step3: Multiply and add (2nd term)

Multiply the result by 6, add to next coefficient.
$(-2) \times 6 = -12$; $11 + (-12) = -1$
$\boldsymbol{-1}$

Step4: Multiply and add (3rd term)

Multiply the result by 6, add to next coefficient.
$(-1) \times 6 = -6$; $6 + (-6) = 0$
$\boldsymbol{0}$

Step5: Multiply and add (4th term)

Multiply the result by 6, add to last coefficient.
$0 \times 6 = 0$; $-3 + 0 = -3$
$\boldsymbol{-3}$

Answer:

(a) Completed synthetic division table (filled values, left to right, top to bottom):

$-2$, $-2$, $-1$, $0$, $-3$

(b)

$\boldsymbol{-2x^3 - 2x^2 - x + 0 + \frac{-3}{x-6}}$
Or simplified:
$\boldsymbol{-2x^3 - 2x^2 - x - \frac{3}{x-6}}$
Quotient: $-2x^3 - 2x^2 - x$
Remainder: $-3$