Factorization by Splitting middle term: Part IV

Factorization by splitting the middle term:- Part IV

There are some expression   which can be factorized by splitting middle term; let us consider this expression

Px2 + qx + r and  Px2 + qxy + ry2

Both are neither a square nor in the term of (x + a)(x + b) = x2 + (a + b)x + ab. Such expressions are factorized by splitting their two middle terms.

viz the middle terms qx of the expression px² + qx + r is split into two terms ax and bx such that their sum is qx and their product is equal to the product of the first and the third term. Let us consider on following example:-

Exp 1: 6x² – 5xy – 6y²

1st step: Get the product of first and third terms

i.e  6x² x 6y² = 36x²y²

2nd step: Split middle terms such as it product would be equal to product of first and third terms.

Here, middle term =  – 5xy

So, we can write it as 4xy – 9xy                     (36x²y²)

Therefore,

6x² – 5xy – 6y² =6x² +4xy – 9xy – 6y²

                         

                            =2x(3x + 2y)(2x – 3y)

Exp 2: 7x² + 23xy + 6y²

Product of the first and the third terms = 42x²Y²

Therefore, the middle term 23xy can be write as

21xy + 2xy

Hence  7x² + 23xy + 6y² = 7x² +21xy + 2xy +6y²

                                  

                                       =7x(x + 3y) + 2y(x + 3y)

                                       = (x +3y)(7x +2y)

Exp 2: 14a² + 11ab – 15b²

Product of the first and the third terms = – 210a²b²

Therefore it can be write as 21ab – 10ab

Hence,  14a² + 11ab – 15b² = 14a²  – 21ab – 10ab – 15b²

                                                 =7a(2a – 3b) – 5b(2a – 3b)

                                                 =(2a – 3b)(7a – 5b)

Exp 4:- x² - 9x + 20 = x²  – 4x – 5x + 20

                                 = x(x – 4) + 5(4 – x)

By taking 5 common from (20 – 5x), we are not getting the expression that is common in both terms. Therefore, we need to take – 5 common from (20 – 5x).

The second step in the above solution will be correctly written as

x²  – 9x + 20 = x(x – 4) – 5(x – 4)

                     = (x – 4)(x – 5)  

Note: it is sometimes necessary to change the order of the terms for getting the expression that is common in terms of the given expression.