Question

Factorize by the grouping method: $16{(a + b)^2} - 4a - 4b$.

Answer 1

Hint: In this question, we will factorize by using a grouping method. Here, we first take 4 common from 4a and 4b, and then we will take (a + b) common. In this way we will further solve this question and represent the final result in the form of product.

Complete step-by-step answer:
We have $16{(a + b)^2} - 4a - 4b$;
Factorizing by using grouping method;
$16{(a + b)^2} - 4a - 4b$
(Taking -4 common from -4a – 4b)
$ = 16{(a + b)^2} - 4(a + b)$
(Here we will take 4(a + b) common)
$ = 4(a + b)\left[ {4(a + b) - 1} \right]$
$ = 4(a + b)\left[ {4a + 4b - 1} \right]$

Therefore $16{(a + b)^2} - 4a - 4b$ = $4(a + b)\left[ {4a + 4b - 1} \right]$

Note: Factorization can be defined as the breaking or decomposing of a number, polynomial or a quadratic equation into a product of another entity, or factors, which when multiplied together give the original number, polynomial or quadratic equation. There are so many general methods to factorize; 1) General method: This method is applied to any expression that is a sum, or that may be transformed into a sum. This method mostly applied to polynomials. 2) Grouping method: Grouping of terms allow using other methods for getting factorization. 3) Common factor method: It may occur that all terms of a sum are products and that some factors are common to all terms. In this case, distributive law allows factoring out this common factor. 4) Adding and subtracting: Sometimes, some term grouping lets a part of a recognizable pattern. In this case it is useful to add terms for completing the pattern, and subtract them for not changing the value of the expression. 5) Recognizable patterns: Many identities provide an equality between a sum and a product.