Question

Factorize:

${(3a-2b)}^3+{(2b-5c)}^3+{(5c-3a)}^3$

Answer 1

Factorizing the given expression:

${(3a-2b)}^3+{(2b-5c)}^3+{(5c-3a)}^3$

Let $A=3a-2b,B=2b-5candC=5c-3a$

$$\begin{gathered} A+B+C=3a-2b+2b-5c+5c-3a \\ A+B+C=0 \end{gathered}$$

Using identity: $A^3+B^3+C^3-3ABC=(A+B+C)(A^2+B^2+C^2-AB-BC-CA)$

So, if $A+B+C=0\Rightarrow A^3+B^3+C^3=3ABC$

$$\begin{gathered} {(3a-2b)}^3+{(2b-5c)}^3+{(5c-3a)}^3 \\ =3(3a-2b)(2b-5c)(5c-3a) \end{gathered}$$

Hence, after factorizing the expression we get, ${(3a-2b)}^3+{(2b-5c)}^3+{(5c-3a)}^3=3(3a-2b)(2b-5c)(5c-3a).$