Question

1. $(x^3 - 13x + 12) div (x + 4)$

Explanation:

Step1: Use polynomial long division

Divide \(x^3 - 13x + 12\) by \(x + 4\). First, divide the leading term \(x^3\) by \(x\) to get \(x^2\). Multiply \(x + 4\) by \(x^2\) to get \(x^3 + 4x^2\). Subtract this from the original polynomial:
\[

$$\begin{align*} &(x^3 - 13x + 12) - (x^3 + 4x^2)\\ =&x^3 - 13x + 12 - x^3 - 4x^2\\ =& -4x^2 - 13x + 12 \end{align*}$$

\]

Step2: Divide the new leading term \(-4x^2\) by \(x\) to get \(-4x\). Multiply \(x + 4\) by \(-4x\) to get \(-4x^2 - 16x\). Subtract this from the previous result:

\[

$$\begin{align*} &(-4x^2 - 13x + 12) - (-4x^2 - 16x)\\ =& -4x^2 - 13x + 12 + 4x^2 + 16x\\ =& 3x + 12 \end{align*}$$

\]

Step3: Divide the new leading term \(3x\) by \(x\) to get \(3\). Multiply \(x + 4\) by \(3\) to get \(3x + 12\). Subtract this from the previous result:

\[

$$\begin{align*} &(3x + 12) - (3x + 12)\\ =& 3x + 12 - 3x - 12\\ =& 0 \end{align*}$$

\]
So, putting it all together, the quotient is \(x^2 - 4x + 3\) and the remainder is \(0\). We can also factor the quotient: \(x^2 - 4x + 3=(x - 1)(x - 3)\).

Answer:

The result of \((x^3 - 13x + 12)\div(x + 4)\) is \(x^2 - 4x + 3\) (or \((x - 1)(x - 3)\)) with a remainder of \(0\).