Question

Resolve into factors: $ 81{a^4} + 9{a^2}{b^2} + {b^4} $

Answer 1

Hint: Here we will use the algebraic identity for $ {x^4} + {y^4} + {x^2}{y^2} $ to factorize the given expression. Even though the degree of the given expression is four, observe that there are no terms whose power is less than four.

Complete step-by-step answer:
The factors of an algebraic expression are terms which when multiplied gives back the expression. For example, the factors of $ 2xy $ are $ 2,x $ and $ y $ . Also, the factors of $ {a^2} - {b^2} $ are $ (a + b)(a - b) $ by using the algebraic identity $ {a^2} - {b^2} = (a + b)(a - b) $ . So, now using this basic understanding of factors we can see that to find the factors of an algebraic expression we have to express it as a product of two or more expressions.
Observe that the degree of the given expression, which is the highest power among all the terms, is four. But even then, we can see that the powers of all the terms are four. Moreover, we can see that the powers of the variables $ a{\text{ and }}b $ are even. So, we will use the algebraic identity for $ {x^4} + {y^4} + {x^2}{y^2} $ to factorize the given expression.
So, the given expression
 $ 81{a^4} + 9{a^2}{b^2} + {b^4} $ - - - - - - - - - - - - - (1)
Can be re-written as
 $ {\left( {3a} \right)^4} + {\left( {3a} \right)^2}{b^2} + {b^4} $
On comparing this expression with $ {x^4} + {y^4} + {x^2}{y^2} $ we get $ x $ to be $ 3a $ and $ y $ to be $ b $ .
Now we will use the identity
 $ {x^4} + {y^4} + {x^2}{y^2} = \left( {{x^2} + {y^2} + xy} \right)\left( {{x^2} + {y^2} - xy} \right) $ to factorize the given expression.
So, we can write
 $ {\left( {3a} \right)^4} + {\left( {3a} \right)^2}{b^2} + {b^4} = \left( {{{\left( {3a} \right)}^2} + {b^2} + 3ab} \right)\left( {{{\left( {3a} \right)}^2} + {b^2} + 3ab} \right) $
 $ \Rightarrow 81{a^4} + 9{a^2}{b^2} + {b^4} = \left( {9{a^2} + {b^2} + 3ab} \right)\left( {9{a^2} + {b^2} - 3ab} \right) $
We cannot further factorize the expressions
 $ \left( {9{a^2} + {b^2} + 3ab} \right) $ and $ \left( {9{a^2} + {b^2} - 3ab} \right) $ .
Hence the factors of the given expression
 $ 81{a^4} + 9{a^2}{b^2} + {b^4} $ is $ \left( {9{a^2} + {b^2} + 3ab} \right) $ and $ \left( {9{a^2} + {b^2} - 3ab} \right) $ .
So, the correct answer is “ $ \left( {9{a^2} + {b^2} + 3ab} \right) $ and $ \left( {9{a^2} + {b^2} - 3ab} \right) $ .”.

Note: The expressions $ \left( {9{a^2} + {b^2} + 3ab} \right) $ and $ \left( {9{a^2} + {b^2} - 3ab} \right) $ are not factorizable because to use the identities $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ and $ {(a - b)^2} = {a^2} + {b^2} - 2ab $ we require an additional 2 multiplied in the $ 3ab $ term of the expressions.