Question 4.3.4: Using Synthetic Division Use synthetic division to divide 2x...
Using Synthetic Division
Use synthetic division to divide 2 x^{4}+x^{3}-16 x^{2}+18 \text { by } x+2 .
STUDY TIP
A quick way to see if you remembered to put 0 coefficients in place of missing terms when dividing a polynomial of degree n by x – a is to check that there are n + 1 numbers in the row next to \underline{a|} .
Because x-a=x+2=x-(-2), we have a=-2. Write the coefficients of the dividend in a line, supplying 0 as the coefficient of the missing x-term. Then carry out the steps of synthetic division.
\begin{matrix} \underline{-2|} &\\ \\ \\ \end{matrix} \begin{matrix} 2 & 1& -16 & 0& 18 \\ & -4&6&20 & -40\\ \hline 2 &-3 & -10 & 20 & |-22 \end{matrix}
The quotient is 2 x^{3}-3 x^{2}-10 x+20, and the remainder is -22; so the result is
\begin{aligned}\frac{2 x^{4}+x^{3}-16 x^{2}+18}{x+2} &=2 x^{3}-3 x^{2}-10 x+20+\frac{-22}{x+2} \\&=2 x^{3}-3 x^{2}-10 x+20-\frac{22}{x+2}.\end{aligned}