Question

Factorise: ${a^{12}} - {b^{12}}$
A. $\left( {{a^6} + {b^6}} \right)\left( {a + b} \right)\left( {{a^2} - ab - {b^2}} \right)\left( {{a^2} +
ab + {b^2}} \right)$
B. $\left( {{a^2} + {b^2}} \right)\left( {{a^4} + {b^4} - {a^2}{b^2}} \right)\left( {a + b} \right)\left(
{{a^2} + {b^2} - ab} \right)\left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$
C. $\left( {{a^6} + {b^6}} \right)\left( {a - b} \right)\left( {{a^2} - ab + {b^2}} \right)\left( {{a^2} +
ab + {b^2}} \right)$
D. $\left( {{a^6} - {b^6}} \right)\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\left( {{a^2} +
ab + {b^2}} \right)$

Answer 1

Hint:In this question, we are supposed to reduce the given algebraic expression into more simple form using identities and formulae such that the final expression cannot be factored further. By using the properties of indices, they convert the equation into identity form. Then, by applying a suitable identity expand the given algebraic expression.

Formulae used:1) ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
2) ${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
3) ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$

Complete step by step solution:
As we can see we have to solve \[{a^{12}} - {b^{12}}\] , but there is no direct formula for solving this so we will try to manipulate the powers of a and b as it to bring it in such an expression which can be simplifies easily.

By looking at this the very first step that we can do is breaking the power 12 as a product of 6 and 2,so
we will get: \[{a^{12}} - {b^{12}}\] = \[{({a^6})^2} - {({b^6})^2}\]
Now we can use the formula (\[{a^2} - {b^2} = (a + b)(a - b)\]), taking a = \[{a^6}\] and b =\[{b^6}\], w
get\[{a^{12}} - {b^{12}} = ({a^6} + {b^6})({a^6} - {b^6})\]

Now it can be further simplified as:
Let us do it step by step, first of all simplify \[({a^6} + {b^6})\]. For this again we will write the power 6 as product of 3 and 2 to set in the cubic formula as shown below
\[{a^{12}} - {b^{12}} = \left[ {{{({a^2})}^3} + {{({b^2})}^3}} \right]\left( {{a^6} - {b^6}} \right)\]
Now using the formula (\[{a^3} + {b^3}\]) = \[(a + b)({a^2} - ab + {b^2})\]taking a = \[{a^2}\]and b
=\[{b^2}\], we get \[
{a^{12}} - {b^{12}} = [({a^2} + {b^2})\{ {({a^2})^2} - {a^2}{b^2} + {({b^2})^2}\} ]({a^6} - {b^6}) \\
= ({a^2} + {b^2})({a^4} - {a^2}{b^2} + {b^4}\} ({a^6} - {b^6}) \\
\]

Now it can’t be further simplified so we will move to simplify \[({a^6} - {b^6})\], first by bring the power to square then to cube as shown in the solution below:\[{a^{12}} - {b^{12}} = \{ ({a^2} + {b^2})({a^4} -
{a^2}{b^2} + {b^4})\} \{ {({a^3})^2} - {({b^3})^2}\} \]
Now using the formula (\[{a^2} - {b^2}\]) = \[(a + b)(a - b)\], taking a = \[{a^3}\]and b = \[{b^3}\], we
get\[{a^{12}} - {b^{12}} = \{ ({a^2} + {b^2})({a^4} - {a^2}{b^2} + {b^4})\} \{ ({a^3} - {b^3})({a^3} +
{b^3})\} \]
Now by using direct formula 2 and 3 last two term can be simplified;\[{a^{12}} - {b^{12}} = \{ ({a^2} +
{b^2})({a^4} - {a^2}{b^2} + {b^4})\} (a - b)({a^2} + ab + {b^2})(a + b)({a^2} - ab + {b^2})\]
Now we get the solution which is\[{a^{12}} - {b^{12}} = ({a^2} + {b^2})({a^4} - {a^2}{b^2} + {b^4})(a -
b)({a^2} + ab + {b^2})(a + b)({a^2} - ab + {b^2})\]

Hence, we get the solution and it matches with option B.

Additional information:Factorization is a method in which the given algebraic expressions are converted to irreducible expression. Irreducible expression is something which cannot be reduced further. Factorization of algebraic expression may give numbers, variables, or number of small algebraic expressions. Factorization is done using various methods/processes which include factorization using common factor, factorization by regrouping, factorization with the help of identities, factorization in the
form of\[\left( {x + a} \right)\left( {x + b} \right)\] .

Note:Indices properties should be applied carefully and properly in order to get the desired answer.
Brackets should be used wherever necessary. Students usually don’t use more than one formula, factoring an equation to its simplest form requires multiple algebraic formulae. Common mistakes are done like not factoring out all common factors or factoring GCF(Greatest Common Factor) from all terms. Or cancelling out a common factor, is also a mistake usually made by students while doing factorization of any equation.