Question

State true or false $\sum a({b^2} - {c^2}) = (a - b)(b - c)(c - a)$

Answer 1

Hint: In order to solve this question we need to first divide the question into two parts Left hand side and Right hand side , to prove that the Left hand side. is equal to Right hand side . Here we are going to expand the Left hand side part and perform some calculations to simplify the given equation equation by somewhere using equivalent equations . Equivalent equations are said to be algebraic equations that may have the same solutions if we add or subtract the same number to both sides of an equation - Left hand side or Right hand side of the equal to sign . Or we can multiply or divide the same number to both sides of an equation - Left hand side or Right hand side of the equal to sign with the method of simplification .

Complete step by step answer:
The question given to us is $\sum a({b^2} - {c^2}) = (a - b)(b - c)(c - a)$ , here first we will consider only left hand side ,
$\sum a({b^2} - {c^2})$
Taking out the summation we get ,
$a{b^2} - a{c^2} + b{c^2} - b{a^2} + c{a^2} - c{b^2}$
Now to make this easier we will perform simplification by making it equivalent equation , we will add and subtract abc at the same time in the equation –
$abc + a{b^2} - a{c^2} + b{c^2} - b{a^2} + c{a^2} - c{b^2} - abc$
Now we will take common a and b both from the equation to make it resembling like right hand side
$a(bc - ab - {c^2} + ac) - b(bc - ab - {c^2} + ac)$
Rewriting and arranging , we get =
$(a - b)(bc - ab - {c^2} + ac)$
Again Rewriting , we get =
$(a - b)(b - c)(c - a)$
Hence , proved .
Thus , the correct answer is True .

Note: In equivalent equations which have identical solutions we can perform multiplication or division by the same non-zero number both L.H.S. and R.H.S. of an equation .
In an equivalent equation which have identical solution we can raise the same odd power to both L.H.S. and R.H.S. of an equation .
Cross check the answer and always keep the final answer simplified .