Question

Factorise $\sum a(b^2-c^2)$

• A. $-(a-b)(b-c)(c-a)$
• B. $(a-b)(b-c)(c-a)$
• C. $(a-b)(b-c)(c+a)$
• D. None

Answer 1

The correct option is B $(a-b)(b-c)(c-a)$

$\sum a(b^2-c^2)=a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$

$\Rightarrow a{b}^2-a{c}^2+(b{c}^2-b{a}^2)+c{a}^2-c{b}^2$

$\Rightarrow (a-c)b^2+ac(a-c)+\left[b\right(c-a\left)\right(c-a\left)\right]$

$\Rightarrow (a-c)(b^2+ac-bc-ab)=(a-c)\left[b\right(b-c)-a(b-c)=(a-b\left)\right(b-c\left)\right(c-a\left)\right]$