Question

How do you factor ${x^2} - 4x - 4{y^2} + 4$ by grouping?

Answer 1

Hint: In this question we have to factorise the polynomial by grouping and we will do this by grouping the like terms which we can easily factor, and by using algebraic identities ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$and $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$ and simplifying the given polynomial, we will get the required factorised term.

Complete step by step solution:
Given polynomial is ${x^2} - 4x - 4{y^2} + 4$,
Now grouping the terms we get,
$ \Rightarrow \left( {{x^2} - 4x + 4} \right) - 4{y^2}$,
Here ${x^2} - 4x + 4$is in form of ${a^2} + 2ab + {b^2}$,
Now using the algebraic identity, ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$, here $a = 1$and $b = 2$now substituting the values in the identity,
$ \Rightarrow {\left( x \right)^2} - 2\left( x \right)\left( 2 \right) + {\left( 2 \right)^2} - 4{y^2}$,
and now write $4{y^2}$as ${\left( {2y} \right)^2}$, we get,
$ \Rightarrow {\left( {x - 2} \right)^2} - {\left( {2y} \right)^2}$,
Now this is in form of algebraic identity $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$, here $a = x - 2$and $b = 2y$ now substituting the values in the identity, we get,
$ \Rightarrow \left( {x - 2 + 2y} \right)\left( {x - 2 - 2y} \right)$,
Now rearranging the terms, we get,
$ \Rightarrow \left( {x + 2y - 2} \right)\left( {x - 2y - 2} \right)$,
So, by factorising the given polynomial we get $\left( {x + 2y - 2} \right)\left( {x - 2y - 2} \right)$.

$\therefore $ The factorising term when the given polynomial ${x^2} - 4x - 4{y^2} + 4$ is factorised will be equal to $\left( {x + 2y - 2} \right)\left( {x - 2y - 2} \right)$.

Note: Factoring by grouping involves grouping terms then factoring out common factors. Factorization is a process which is necessary to simplify the algebraic expressions and is used to solve the higher degree equations. It is the inverse procedure of the multiplication of the polynomials. The algebraic expression is said to be in a factored form only when the whole expression is an indicated product. Factorisation of the algebraic expression can also be done in other methods such as, factorising by grouping and factorising using identities. Algebraic identities are algebraic equations which are always true for every value of variables in them. In an algebraic identity, the left-side of the equation is equal to the right-side of the equation, some of the algebraic identities that are commonly used are:
${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$,
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$,
$\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$,
${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$,
${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$.