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

State true or false ∑a(b2−c2)=(a−b)(b−c)(c−a)\sum a({b^2} - {c^2}) = (a - b)(b - c)(c - a)

Explanation

Solution

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 ∑a(b2−c2)=(a−b)(b−c)(c−a)\sum a({b^2} - {c^2}) = (a - b)(b - c)(c - a) , here first we will consider only left hand side ,
∑a(b2−c2)\sum a({b^2} - {c^2})
Taking out the summation we get ,
ab2−ac2+bc2−ba2+ca2−cb2a{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+ab2−ac2+bc2−ba2+ca2−cb2−abcabc + 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−c2+ac)−b(bc−ab−c2+ac)a(bc - ab - {c^2} + ac) - b(bc - ab - {c^2} + ac)
Rewriting and arranging , we get =
(a−b)(bc−ab−c2+ac)(a - b)(bc - ab - {c^2} + ac)
Again Rewriting , we get =
(a−b)(b−c)(c−a)(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 .