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Factors of the Form (x + a)(x + b)

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Factors of the form (x + a)(x + b):

In general, for factorising an algebraic expression of the type x2 + px + q, we find two factors a, and b of q (i.e., the constant term) such that

ab = q and a + b = p

Then, the expression becomes x2 + (a + b) x + ab

or   x2 + ax + bx + ab

or   x(x + a) + b(x + a)

or   (x + a)(x + b) which are the required factors.

Example

Find the factors of y2 – 7y + 12.
We note 12 = 3 × 4 and 3 + 4 = 7.
Therefore,
y2 – 7y + 12 = y2 – 3y – 4y + 12
= y(y – 3) – 4 (y – 3)
= (y – 3)(y – 4)

Example

Obtain the factors of z2 – 4z – 12.

z2 – 4z – 12 
= z2 – 6z + 2z – 12 
= z(z - 6) + 2(z - 6) 
= (z - 6)(z + 2)

Example

Find the factors of 3m2 + 9m + 6.
We notice that 3 is a common factor of all the terms.
Therefore,
3m2 + 9m + 6 = 3(m2 + 3m + 2)
(m2 + 3m + 2) = m2 + m + 2m + 2          ......(as 2 = 1 × 2)
                        = m(m + 1) + 2(m + 1)
                        = (m + 1)(m + 2)
3m2 + 9m + 6 = 3(m + 1)(m + 2).
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Shaalaa.com | Factors of the Form (x + a)(x + b) - Splitting the middle term

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