Question

rewrite the rational expression \\(\frac{x^2 + 2}{x - 2}\\) in the form \\(q(x) + \frac{r(x)}{b(x)}\\). use long division method to rewrite the rational expression. use the keyboard to type your answer in the box.

Explanation:

Step1: Divide leading terms

Divide $x^3$ by $x$: $\frac{x^3}{x}=x^2$

Step2: Multiply divisor by result

Multiply $x-2$ by $x^2$: $x^2(x-2)=x^3-2x^2$

Step3: Subtract from dividend

Subtract from $x^3+2$:
$(x^3+2)-(x^3-2x^2)=2x^2+2$

Step4: Divide new leading terms

Divide $2x^2$ by $x$: $\frac{2x^2}{x}=2x$

Step5: Multiply divisor by result

Multiply $x-2$ by $2x$: $2x(x-2)=2x^2-4x$

Step6: Subtract from new dividend

Subtract from $2x^2+2$:
$(2x^2+2)-(2x^2-4x)=4x+2$

Step7: Divide new leading terms

Divide $4x$ by $x$: $\frac{4x}{x}=4$

Step8: Multiply divisor by result

Multiply $x-2$ by $4$: $4(x-2)=4x-8$

Step9: Subtract to find remainder

Subtract from $4x+2$:
$(4x+2)-(4x-8)=10$

Answer:

$x^2+2x+4+\frac{10}{x-2}$