Question

use long division to rewrite this expression in an equivalent form as $q(x) + \frac{r(x)}{b(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor. $x^3 - 2x^2 - 4 \div (x - 3)$ drag the expressions to the correct locations on the image. not all expressions will be used. for help, see this worked example. -13 5 $x^2 - 5x + 15$ $x^2 + 5x - 15$ $x^2 + x + 3$ $x - 3$

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(x^3 - 2x^2 - 4\) (which is \(x^3\)) by the leading term of the divisor \(x - 3\) (which is \(x\)). So, \(\frac{x^3}{x}=x^2\). This is the first term of the quotient \(q(x)\).

Step2: Multiply and subtract

Multiply the divisor \(x - 3\) by \(x^2\) to get \(x^3 - 3x^2\). Subtract this from the dividend:
\[

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

\]

Step3: Divide the new leading term

Now, divide the leading term of the new dividend \(x^2 - 4\) (which is \(x^2\)) by the leading term of the divisor \(x\). So, \(\frac{x^2}{x}=x\). Add this to the quotient.

Step4: Multiply and subtract again

Multiply the divisor \(x - 3\) by \(x\) to get \(x^2 - 3x\). Subtract this from the new dividend \(x^2 - 4\):
\[

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

\]

Step5: Divide the new leading term again

Divide the leading term of the new dividend \(3x - 4\) (which is \(3x\)) by the leading term of the divisor \(x\). So, \(\frac{3x}{x}=3\). Add this to the quotient.

Step6: Multiply and find the remainder

Multiply the divisor \(x - 3\) by \(3\) to get \(3x - 9\). Subtract this from the new dividend \(3x - 4\):
\[

$$\begin{align*} &(3x - 4)-(3x - 9)\\ =&3x - 4 - 3x + 9\\ =&5 \end{align*}$$

\]
So, the quotient \(q(x)=x^2 + x + 3\), the remainder \(r(x) = 5\), and the divisor \(b(x)=x - 3\).

Answer:

The first box (quotient) should have \(x^2 + x + 3\), the numerator of the fraction should have \(5\), and the denominator of the fraction should have \(x - 3\). So the filled form is:

\(x^2 + x + 3\) \(+\) \(\frac{5}{x - 3}\)