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Home >> Polynomials >> Factoring of Quadratic Polynomials >>

 Factoring Quadratic Polynomials by Splitting Middle Term

What are Polynomials ?

There are 3 formulas to factor a quadratic polynomial

1) (a + b)2 = a2 + 2ab + b2
2) (a - b)2 = a2 - 2ab + b2
3) a2 - b2 = (a + b) (a - b)

To factorize a given polynomial using the above formulas we will first check that whether we can square the first and last term, if we cannot then instead of these formulas we will use Splitting the Middle Term formula for factorization

Let's first study some examples on these 3 formulas

Example : 4z2 + 12z + 9
Solution : As we can square the first and last term, and all arithmetic signs are of addition we will use the formula (a + b)2 = a2 + 2ab + b2 and we get

Squaring first term 4z2 = (2z)2
Squaring last term 9 = (3)2

(a + b)2 = (2z)2 + 2 X 2z X 3 + (3)2
(a + b)2 = (2z + 3)2

Example : 4z2 - 12z + 9
Solution : As we can square the first and last term, and arithmetic sign of middle term is of subtraction we will use the formula (a - b)2 = a2 - 2ab + b2 and we get

Squaring first term 4z2 = (2z)2
Squaring last term 9 = (3)2

(a - b)2 = (2z)2 - 2 X 2z X 3 + (3)2
(a - b)2 = (2z - 3)2

Example : 4z2 - 9
Solution : As this polynomial has 2 terms and we can square the first and last term and arithmetic sign is of subtraction we will use the formula a2 - b2 = (a + b) (a - b) and we get

Squaring 4z2 = (2z)2
Squaring term 9 = (3)2

a2 - b2 = (2z)2 - (3)2
a2 - b2 = (2z - 3) (2z + 3)