Question

divide using synthetic division.
\\((x^4 - 12x^2 - 11) \div (x + 3)\\)

the quotient is \\(\square\\) and the remainder is \\(\square\\).
(simplify your answers. do not factor.)

Explanation:

Step1: Identify root for divisor

For $x+3=0$, root is $x=-3$
List coefficients of dividend $x^4 + 0x^3 -12x^2 +0x -11$: $1, 0, -12, 0, -11$

Step2: Set up synthetic division

Bring down leading coefficient:

$$\begin{array}{r|rrrrr} -3 & 1 & 0 & -12 & 0 & -11 \\ & & -3 & 9 & 9 & -27 \\ \hline & 1 & -3 & -3 & 9 & -38 \\ \end{array}$$

Step3: Derive quotient and remainder

Quotient uses first 4 results as coefficients of $x^3$ to constant term: $x^3 -3x^2 -3x +9$
Final value is remainder: $-38$

Answer:

The quotient is $x^3 - 3x^2 - 3x + 9$ and the remainder is $-38$.