Question

Resolve into factors: $9a^4+100b^4+30a^2{b}^2$

Answer 1

We know the identity $a^4+b^4+a^2{b}^2=(a^2+b^2+ab)(a^2+b^2-ab)$

Using the above identity, the equation $9a^4+100b^4+30a^2{b}^2$ can be factorised as follows:

$9a^4+100b^4+30a^2{b}^2=(\sqrt{3}a{)}^4+(\sqrt{10}b{)}^4+\left(\sqrt{3}a{)}^2\right(\sqrt{10}b{)}^2$

$=\left[\right(\sqrt{3}a{)}^2+(\sqrt{10}b{)}^2+(\sqrt{3}a\left)\right(\sqrt{10}b\left)\right]\left[\right(\sqrt{3}a{)}^2+(\sqrt{10}b{)}^2+(\sqrt{3}a\left)\right(\sqrt{10}b\left)\right]$

$=(3a^2+10b^2+\sqrt{30}ab)(3a^2+10b^2-\sqrt{30}ab)$

Hence, $9a^4+100b^4+30a^2{b}^2=(3a^2+10b^2+\sqrt{30}ab)(3a^2+10b^2-\sqrt{30}ab)$