Question

factorise: a^4+a^2b^2+b^4

Answer 1

a^4+a^2b^2+b^4
= (a^2+b^2)^2. [since , by using first identity which states that a^2+b^2=(a+b)^2]

Answer 2

The given expression is:

$a^4$ + $a^2b^2$ + $b^4$

We have to find the factorisation of the given expression.

Solution:

∴ $a^4$ + $a^2b^2$ + $b^4$

= $(a^2)^2$ + $(b^2)^2$ + $a^2b^2$

Using the trigonometric identity:

$(x+y)^{2} =x^{2} +y^{2}$ + 2xy

⇒ $x^{2} +y^{2}$ = $(x+y)^{2}$ - 2xy

= $(a^2+b^2)^2$ - 2$a^2b^2$  + $a^2b^2$

= $(a^2+b^2)^2$ - $a^2b^2$  

= $(a^2+b^2)^2$ - $(ab)^2$  

Using the trigonometric identity:

$x^{2} -y^{2}$ = (x + y)(x - y)

= ($a^2+b^2$ + ab)($a^2+b^2$ - ab)

∴ $a^4$ + $a^2b^2$ + $b^4$ = ($a^2+b^2$ + ab)($a^2+b^2$ - ab)

Thus, the factorisation of the given expression is

"($a^2+b^2$ + ab)($a^2+b^2$ - ab)".