Question

use synthetic division to find ((9x^{4}+33x^{3}-11x^{2}+12x + 29)div(x + 4)). write your answer in the form (q(x)+\frac{r}{d(x)}), where (q(x)) is a polynomial, (r) is an integer, and (d(x)) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify root of divisor

For $x+4=0$, root is $x=-4$.
Coefficients of dividend: $9, 33, -11, 12, 29$

Step2: Set up synthetic division

-4 |  9   33   -11   12   29
     |     -36    12   -4   -32
     --------------------------
       9   -3     1    8   -3

Step3: Form quotient and remainder

Quotient $q(x)$: $9x^3 - 3x^2 + x + 8$
Remainder $r$: $-3$
Divisor $d(x)$: $x+4$

Answer:

$9x^3 - 3x^2 + x + 8 + \frac{-3}{x+4}$