Question

Factorize the following expression:

$a^4-16(b-c{)}^4$

Answer 1

Given: $a^4-16(b-c{)}^4$

$\therefore a^4-16(b-c{)}^4$
$=(a^2{)}^2-[4(b-c{)}^2{]}^2$

Using the formula, $(x^2-y^2)=(x-y)(x+y)$
$=[a^2+4(b-c{)}^2[a^2-4(b-c{)}^2]$
$=[a^2+4(b-c{)}^2[a^2-[2(b-c){]}^2]$
​​​​​​​$=[a^2+4(b-c{)}^2\left]\right[a+2(b-c)\left]\right[a-2(b-c)]$
​​​​​​​$=[a^2+4(b-c{)}^2\left]\right(a+2b-2c\left)\right(a-2b+2c)$

Hence, $a^4-16(b-c{)}^4=[a^2+4(b-c{)}^2](a+2b-2c)(a-2b+2c).$