Factorization by Middle Term Splitting


 
 
Concept Explanation
 

Factorization by Middle Term Splitting

Factorization by Middle Term Splitting:

Factorization of quadratic ploynomials in one Variables:

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

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

Clearly, such factos 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.

Example:  Factorize each of the following expressions:

                (i);;x^{2}+6x+8     (ii);;x^{2}+4x-21   (iii);;x^{2}-7x+12

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

                     p + q + 6 and pq = 8

                    Clealry, 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)

(ii)   In order to factorize x^{2}+4x-21, we have to find two numbers p and q such that

     p + q = 4  and  pq = -21

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

     We now split the middle term 4x of x^{2}+4x-21  as  7x - 3x

     therefore ;;x^{2}+4x-21 = x^{2}+7x-3x-21

                                        =(x^{2}+7x)-(3x+21)

                                        =x(x+7)-3(x+7)

                                        =(x+7)(x-3)

(iii)  In order to factorize  large dpi{100} fn_jvn x^{2}-7x+10  we have to find two numbers p and q such that

      p + q = -7 and pq = 10

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

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

      large therefore ;;x^{2}-7x+10=x^{2}-5x-2x+10

                                          large =(x^{2}-5x)-(2x-10)

                                          large =x(x-5)-2(x-5)

                                          large =(x-5)(x-2)

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Express the following as the product of two factors: large 4x^2+12x+9

Right Option : C
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Explanation
Question : 2

The factors of  3m^2+9m+6 are not equal to

Right Option : D
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Explanation
Question : 3

Which of the following values is a zero of the polynomial x^2+3x+2  ?

Right Option : A
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Explanation
 
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