Question

Factorize the expression and divide them as directed:
$(x^4-16)÷x^3+2x^2+4x+8$

Answer 1

Step 1: Factorize the expression

Numeator : $x^4-16$
$={\left(x^2\right)}^2-4^2$

$=(x^2-4)(x^2+4)[\because a^2-b^2=(a+b)(a-b)]$

$=(x^2-2^2)(x^2+4)$

$=(x-2)(x+2)(x^2+4)$

$[\therefore a^2-b^2=(a+b)(a-b)]$

Denominator: $x^3+2x^2+4x+8$

$=x^2(x+2)+4(x+2)(x^2+8)+(2x^2+4x)$

$=(x+2)(x^2+4).(x^3+8)+2x(x+2)$


Step 2: Perform division of given expression

$\frac{x^4-16}{x^3+2x^2+4x+8}=\frac{(x-2)(x+2)(x^2+4)}{(x+2)(x^2+4)}$

$=x-2$

$\therefore (x^4-16)÷x^3+2x^2+4x+8=x-2$.