Question

use synthetic division to simplify the following.
$(h^{2}+h-12)(h-3)^{-1} = \square$

Explanation:

Step1: Rewrite as division problem

The expression $(h^2 + h - 12)(h-3)^{-1}$ is equivalent to $\frac{h^2 + h - 12}{h-3}$, so we divide $h^2 + h - 12$ by $h-3$ using synthetic division. First, identify the root of the divisor: $h-3=0 \implies h=3$. List coefficients of the dividend: $1, 1, -12$.

Step2: Set up synthetic division

Write the root 3, then the coefficients:

$$\begin{array}{r|rrr} 3 & 1 & 1 & -12 \\ & & 3 & 12 \\ \hline & 1 & 4 & 0 \end{array}$$

Step3: Interpret the result

The first two numbers in the bottom row are coefficients of the quotient polynomial, and the last number is the remainder. The remainder is 0, so the quotient is $1h + 4$.

Answer:

$h + 4$ (where $h
eq 3$)