Question

use synthetic

1. $(n^3 - 4n^2 + 9) div (n - 4)$

Explanation:

Step1: Identify root & coefficients

Root of divisor $n-4$ is $4$. Coefficients of dividend $n^3-4n^2+0n+9$: $1, -4, 0, 9$

Step2: Bring down leading coefficient

$$\begin{array}{r|rrrr}4&1&-4&0&9\\&&&&\\\hline&1&&& \end{array}$$

Step3: Multiply & add column 1

Multiply $1\times4=4$, add to $-4$: $-4+4=0$

$$\begin{array}{r|rrrr}4&1&-4&0&9\\&&4&&\\\hline&1&0&& \end{array}$$

Step4: Multiply & add column 2

Multiply $0\times4=0$, add to $0$: $0+0=0$

$$\begin{array}{r|rrrr}4&1&-4&0&9\\&&4&0&\\\hline&1&0&0& \end{array}$$

Step5: Multiply & add column 3

Multiply $0\times4=0$, add to $9$: $9+0=9$

$$\begin{array}{r|rrrr}4&1&-4&0&9\\&&4&0&0\\\hline&1&0&0&9 \end{array}$$

Step6: Write quotient & remainder

Coefficients $1,0,0$ give $n^2+0n+0=n^2$, remainder is $9$.

Answer:

$n^2 + \frac{9}{n-4}$