Question

Find the factors of $a\left(b^4-c^4\right)+b\left(c^4-a^4\right)+c\left(a^4-b^4\right).$

Answer 1

Hint: Try to use the formula of $\left(a^2-b^2\right)=\left(a+b\right)\left(a-b\right)$.

We can write above equation as $a\left(b^4-c^4\right)-b\left(b^4-c^4\right)-b\left(a^4-b^4\right)+c\left(a^4-b^4\right).$
$\Rightarrow \left(b^2-c^2\right)\left(b^2+c^2\right)\left(a-b\right)+\left(a^2-b^2\right)\left(a^2+b^2\right)\left(c-b\right)$
$\Rightarrow \left(b-c\right)\left(b+c\right)\left(b^2+c^2\right)\left(a-b\right)+\left(a-b\right)\left(a+b\right)\left(a^2+b^2\right)\left(c-b\right)$
$\Rightarrow \left(b-c\right)\left(a-b\right)\left(\left(b+c\right)\left(b^2+c^2\right)-\left(a+b\right)\left(a^2+b^2\right)\right)$
$\Rightarrow \left(b-c\right)\left(a-b\right)\left(b^3+bc+c{b}^2+c^3-a^3-a{b}^2-b{a}^2-b^3\right)$
$\Rightarrow \left(b-c\right)\left(a-b\right)\left(bc\left(c+b\right)-ab\left(a+b\right)+\left(c-a\right)\left(c^2+a^2+ac\right)\right)$
$\Rightarrow \left(b-c\right)\left(a-b\right)\left(b\left(c^2+cb-a^2-ab\right)+\left(c-a\right)\left(c^2+a^2+ac\right)\right)$
$\Rightarrow \left(b-c\right)\left(a-b\right)\left(b\left(c^2-a^2\right)+b^2\left(c-a\right)\right)+\left(c-a\right)\left(c^2+a^2+ac\right)$
$\Rightarrow \left(b-c\right)\left(a-b\right)\left(c-a\right)\left(b\left(c+a\right)+b^2+c^2+a^2+ac\right)$
Answer $\Rightarrow \left(b-c\right)\left(a-b\right)\left(c-a\right)\left(a^2+b^2+c^2+ac+bc+ab\right)$

Note: In these questions, try to solve the terms one by one by taking common out and using formulas to simplify.