Factorization by Middle Term Splitting

Factorization by Middle Term Splitting:

Factorization of quadratic ploynomials in one Variables:

Let us now consider the polynomial .

Compairing this with polynomial , 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 abd ab is positive. Therefore, a and b both must negative

Clearly, such factos are -3 and -4.

Hence, the factors of   are (y - 3) and ( y - 4 ).

Algorithm:

Step I  Obtain the quadratic polynomial .

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.

Example:  Factorize each of the following expressions:

Solution    (i)  In order to factorize  , we find two numbers p and q such that

                     p + q + 6 and pq = 8

                    Clealry, 2 + 4 = 6 and 2 4 = 8.

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

(ii)   In order to factorize , we have to find two numbers p and q such that

     p + q = 4  and  pq = -21

      Clearly, 7 + (-3) = 4  and  7 -3 = -21

     We now split the middle term 4x of   as  7x - 3x

(iii)  In order to factorize    we have to find two numbers p and q such that

      p + q = -7 and pq = 10

      Clearly, - 5- 2 = -7 and -5 -2 = 10

      We now split the middle term -7x of the given quadratic as -2x - 5x