Question

Simplify the following two polynomials in the simplest form: $\frac{x^3}{x-2}+\frac{8}{2-x}$

• A. $x^2+2x+4$
• B. $x^2-2x+4$
• C. $x^2+2x-4$
• D. None of these

Answer 1

The correct option is A $x^2+2x+4$

The given expression $\frac{x^3}{x-2}+\frac{8}{2-x}$can be simplified as shown below:

$=\frac{x^3}{x-2}+\frac{8}{2-x}$

$=\frac{x^3}{x-2}+\frac{8}{-(x-2)}$

$=\frac{x^3}{x-2}-\frac{8}{x-2}$

$=\frac{x^3-8}{x-2}$

$=\frac{x^3-2^3}{x-2}$

$=\frac{(x-2)(x^2+(x\times 2)+2^2)}{x-2}(\because a^3-b^3=(a-b\left)\right(a^2+ab+b^2\left)\right)$

$=\frac{(x-2)(x^2+2x+4)}{(x-2)}$

$=x^2+2x+4$

Hence, $\frac{x^3}{x-2}+\frac{8}{2-x}=x^2+2x+4$.