Polymomial ond Rational Functions
Symthotic division

Use synthetic division to find the quotient and remainder when $x^{3}+6 x^{2}-8 x+2$ is divided by $x+7$ by completing the parts below.
(a) Complete this synthetic division table.
(b) Write your answer in the following form: Quotient $+\frac{\text { Remainder }}{x+7}$.
\[
\frac{x^{3}+6 x^{2}-8 x+2}{x+7}=\square+\frac{\square}{x+7}
\]

Step 1 ：Write down the coefficients of the polynomial we're dividing, which are 1, 6, -8, and 2. Also write down the value that makes the divisor equal to zero, which is -7 in this case. The synthetic division table looks like this: \[\begin{array}{r|rrrr} -7 & 1 & 6 & -8 & 2 \\ & & -7 & 7 & -7 \\ \hline & 1 & -1 & -1 & -5 \end{array}\]

Step 2 ：The numbers on the bottom row are the coefficients of the quotient and the remainder.

Step 3 ：The quotient is \(x^{2} - x - 1\) and the remainder is -5. So, the division can be written as: \[\frac{x^{3}+6 x^{2}-8 x+2}{x+7}=x^{2} - x - 1+\frac{-5}{x+7}\]

Step 4 ：The final answer is \[\boxed{\frac{x^{3}+6 x^{2}-8 x+2}{x+7}=x^{2} - x - 1+\frac{-5}{x+7}}\]