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Question: Factorize \({x^2} + 4x - 21\)....

Factorize x2+4x−21{x^2} + 4x - 21.

Explanation

Solution

Factors are numbers or algebraic expressions that divide another number or algebraic expression completely without leaving any remainder and factorization is the process of breaking the polynomial entity into its factors. One such method for factorization is the Grouping method.

Grouping method: In this method we work upon the expression which is of this form x2+ax+b{x^2} + ax + band then look for two factors p  &  qp\;\& \;q such that
p×q=b   &  p+q=ap \times q = b\,\;\& \;p + q = a
So here for the given polynomial we can perform factorization by grouping method and thus find its
factors.

Complete step by step solution:
Given
x2+4x−21.....................(i){x^2} + 4x - 21.....................\left( i \right)
Now in order to factorize it we use the Grouping method.
As mentioned above Grouping method: In this method we work upon the expression which is of this form x2+ax+b{x^2} + ax + b and then look for two factors p  &  qp\;\& \;q such that
p×q=b   &  p+q=a....................(ii)p \times q = b\,\;\& \;p + q = a....................\left( {ii} \right)
Then the given expression can be written as
x2+(a+b)x+ab{x^2} + (a + b)x + ab
Multiply xx inside (a+b)\left( {a + b} \right), we get:
x2+ax+bx+ab ⇒x(x+a)+b(x+a) ⇒(x+a)(x+b)  {x^2} + ax + bx + ab \\\ \Rightarrow x(x + a) + b(x + a) \\\ \Rightarrow (x + a)(x + b) \\\
These expressions, (x+a)  ,(x+b)(x + a)\;,(x + b) are the factors of the given algebraic expression.
Now on comparing with (i) and (ii) we can write p=7p = 7 and q=−3q = - 3. Thus:
7×−3=21=b 7+(−3)=4=a  7 \times - 3 = 21 = b \\\ 7 + \left( { - 3} \right) = 4 = a \\\
Now on using Grouping Method we can write:

x2+4x−21 =  x2+(7−3)x+7×−3 =x2+7x−3x−21.......................(iii) {x^2} + 4x - 21 \\\ = \;{x^2} + (7 - 3)x + 7 \times - 3 \\\ = {x^2} + 7x - 3x - 21.......................\left( {iii} \right) \\\

we have to take the common terms from each of the terms.
=  x(x+7)−3(x+7) =(x+7)(x−3)  = \;x(x + 7) - 3(x + 7) \\\ = (x + 7)(x - 3) \\\
Therefore on factoring x2+4x−21{x^2} + 4x - 21 we get the factors as (x+7)   and    (x−3)(x + 7)\,\;and\;\;(x - 3) .

Note: The Grouping method is used when the algebraic expression given is in the form where the 2 nd term and 3 rd can be split into products and sums as shown above. If the given algebraic expression is in the form of any known identity then we can perform factorization easily using conventional methods or else methods like grouping are used. In grouping methods care should be taken while choosing p  and  qp\;and\;qsince it is an area where human errors can interfere.