Question

do now
evaluate $x^3+11x^2+37x+42$ using synthetic division
$x+6$
is $(x+6)$ a factor of the numerator function. explain

Explanation:

Step1: Identify root for divisor

For $x+6=0$, root is $x=-6$.
Coefficients of dividend: $1, 11, 37, 42$

Step2: Set up synthetic division

$$\begin{array}{r|rrrr} -6 & 1 & 11 & 37 & 42 \\ \hline & & -6 & -30 & -42 \\ \hline & 1 & 5 & 7 & 0 \end{array}$$

Step3: Interpret remainder and quotient

Remainder = 0, so $x+6$ is a factor.
Quotient: $x^2 + 5x + 7$

Answer:

The quotient is $x^2 + 5x + 7$, and $x+6$ is a factor of $x^3+11x^2+37x+42$ because the remainder from synthetic division is 0.