Factorization by Middle Term Splitting


 
 
Concept Explanation
 

Factorization by Middle Term Splitting

Factorization of quadratic polynomials in one Variables:

Let us now consider the polynomial y^{2}-7y+12.

Comparing this with polynomial y^{2}+(a+b)y+ab, we get a + b = -7 and ab = 12.

Now, we have to find factors of 12 whose sum is -7.

Since a + b is negative  ab is positive. Therefore, a and b both must negative

Clearly, such factors are -3 and -4.

Hence, the factors of y^{2}-7y+12  are (y - 3) and ( y - 4 ).

therefore ;;y^{2}-7y+12=(y-3)(y-4).

Algorithm:

Step I  Obtain the quadratic polynomial x^{2}+px+q.

Step II  Obtain p = coefficient of x and, q = constant term.

Step III  Find two numbers a and b such that a + b = p and ab = q.

Step IV  Split up the middle term as the sum of two terms ax and bx.

Step V  Factorize the expression obtained in step IV by grouping the terms.

Illustration:  Factorize the following expression:

                ;;x^{2}+6x+8    

Solution:  In order to factorize  x^{2}+6x+8, we find two numbers p and q such that

                     p + q + 6 and pq = 8

                    Clearly, 2 + 4 = 6 and 2 times 4 = 8.

                   We now split the middle term 6x in the given quadratic as 2x + 4x.

                   therefore ;;;x^{2}+6x+8=x^{2}+2x+4x+8

                                                     =(x^{2}+2x)+4(4x+8)

                                                    =x(x+2)+4(x+2)

                                                    =(x+2)(x+4)

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Sample Questions
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Question : 1

The sum and the difference of two expressions are  5x^2-x-4 ; and ; x^2+9x-10  respectively, then their LCM would be equal to :

Right Option : D
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Explanation
Question : 2

Factorize:   z^2-12z+27

Right Option : B
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Explanation
Question : 3

Solve    x^{2}-9x+20=0

Right Option : A
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Explanation
 
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