Question

Factorise $81-x^4$ is ____

• A. $(3-x)(x+3)(x^2+9)$
• B. $(3-2x)(2x+3)(x^2+9)$
• C. $(x-3)(x+3)(2x^2+9)$
• D. $(x-3)(x+3)(x^2+3)$

Answer 1

The correct option is A $(3-x)(x+3)(x^2+9)$

$81-x^4$
We know $a^2-b^2=(a+b)(a-b)$
$81-x^4=(3^2{)}^2-(x^2{)}^2$
Now, let $x^2=k$
$(3^2{)}^2-(x^2{)}^2=(9{)}^2-(k{)}^2=(k+9)(9-k))$
$(k+9)(9-k)=(x^2+3^2)(3^2-x^2)$
Factorising further , we have
$(x^2+3^2)(3^2-x^2)=(x^2+9)(3+x)(3-x)$