Question

Evaluate the following using the identities, \[\left( {{a^2} + {b^2}} \right) \times \left( { - {a^2} + {b^2}} \right)\] ?
a) $\left(b^3-a^3\right)$
b) $\left(b^4-a^4\right)$
c) $\left(b^3+a^3\right)$
d) $\left(b^4+a^4\right)$

Answer 1

Hint: We know that algebraic identities are algebraic equations that are valid for all values of variables in them. We can do Polynomial factorization with them. Algebraic identities are utilized in this way to compute algebraic expressions and solve various polynomials. Unlike algebraic expressions, algebraic identities satisfy all the values of the variables.

Complete step-by-step solution:
We have to simplify the expression
\[\left( {{a^2} + {b^2}} \right) \times \left( { - {a^2} + {b^2}} \right)\]
We will use the identity
\[\left( {x + y} \right) \times \left( {x - y} \right) = {x^2} - {y^{2\;\;}}\] (1)
we can write equation \[\left( {{a^2} + {b^2}} \right) \times \left( { - {a^2} + {b^2}} \right)\] also as \[\left( {{b^2} + {a^2}} \right) \times \left( {{b^2} - {a^2}} \right)\]
so,
\[\left( {{a^2} + {b^2}} \right) \times \left( { - {a^2} + {b^2}} \right) = \left( {{b^2} + {a^2}} \right) \times \left( {{b^2} - {a^2}} \right)\]
so, we will now compare equation \[\left( {{b^2} + {a^2}} \right) \times \left( {{b^2} - {a^2}} \right)\]with equation 1,
we get that here \[{b^2} = x\] and \[{a^2} = y\] ,
$ = \left( {{b^2} + {a^2}} \right) \times \left( {{b^2} - {a^2}} \right)$
$ = \left( {{b^4} - {a^4}} \right)$
We get that equation \[\left( {{a^2} + {b^2}} \right) \times \left( { - {a^2} + {b^2}} \right)\] is identical as $\left( {{b^4} - {a^4}} \right)$.

So, our correct option is b.

Additional information: A variable and constant expression is known as an algebraic expression. A variable in an expression can have any value. As a result, if the variable values change, the expression value can change. Algebraic identity, on the other hand, is equality that holds for all values of the variables.

Note: We can use the binomial theorem to deduce all algebraic identities. Algebraic identities are extremely useful for factoring algebraic expressions quickly. We should know some identities like \[{a^2} - {b^2} = \left( {a - b} \right) \times (a + b)\] , \[{a^3} + {b^3}\; = \left( {a + b} \right) \times \left( {{a^2} - ab + {b^2}} \right)\] are important identities.