Question

divide using polynomial long division or synthetic division.\\((x^4 - x^2 - 7) \div (x + 4) = \square\\)

Explanation:

Step1: Set up synthetic division

Root of divisor $x+4$ is $-4$.
Coefficients of dividend: $1, 0, -1, 0, -7$ (for $x^4+0x^3-x^2+0x-7$)

Step2: Bring down leading coefficient

$$\begin{array}{r|rrrrr} -4 & 1 & 0 & -1 & 0 & -7 \\ \hline & & & & & \\ \hline & 1 & & & & \end{array}$$

Step3: Multiply and add (1st pass)

Multiply $1\times(-4)=-4$, add to next coefficient: $0+(-4)=-4$

$$\begin{array}{r|rrrrr} -4 & 1 & 0 & -1 & 0 & -7 \\ \hline & & -4 & & & \\ \hline & 1 & -4 & & & \end{array}$$

Step4: Multiply and add (2nd pass)

Multiply $-4\times(-4)=16$, add to next coefficient: $-1+16=15$

$$\begin{array}{r|rrrrr} -4 & 1 & 0 & -1 & 0 & -7 \\ \hline & & -4 & 16 & & \\ \hline & 1 & -4 & 15 & & \end{array}$$

Step5: Multiply and add (3rd pass)

Multiply $15\times(-4)=-60$, add to next coefficient: $0+(-60)=-60$

$$\begin{array}{r|rrrrr} -4 & 1 & 0 & -1 & 0 & -7 \\ \hline & & -4 & 16 & -60 & \\ \hline & 1 & -4 & 15 & -60 & \end{array}$$

Step6: Multiply and add (4th pass)

Multiply $-60\times(-4)=240$, add to last coefficient: $-7+240=233$

$$\begin{array}{r|rrrrr} -4 & 1 & 0 & -1 & 0 & -7 \\ \hline & & -4 & 16 & -60 & 240 \\ \hline & 1 & -4 & 15 & -60 & 233 \end{array}$$

Step7: Write quotient and remainder

Coefficients $1,-4,15,-60$ form quotient $x^3-4x^2+15x-60$, remainder $233$.

Answer:

$x^3 - 4x^2 + 15x - 60 + \frac{233}{x+4}$