Factoring of Quadratic Polynomials : Solved Examples

Factoring Quadratic Polynomials by Splitting Middle Term

Factorize the following - A) 9y2 + 24y + 16 B) 9x2 - 16 C) 4y2 - 12y + 9 D) 144x2 + 24x + 1 E) 36y2 - 60y + 25 F) 16x2 - 64 G) x3 - x H) 4x2 - (2y - z)2 A) 9y2 + 24y + 16 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 9y2 = (3y)2 Squaring last term 16 = (4)2 (a + b)2 = (3x)2 + 2 X 3y X 4 + (4)2 (a + b)2 = (3x + 4)2 B) 9x2 - 16 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 term 9x2 = (3x)2 Squaring term 16 = (4)2 a2 - b2 = (3x)2 - (4)2 a2 - b2 = (3x + 4) (3x - 4) C) 4y2 - 12y + 9 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 4y2 = (2y)2 Squaring last term 9 = (3)2 (a - b)2 = (2y)2 + 2 X 2y X 3 + (3)2 (a - b)2 = (2y - 3)2 D) 144x2 + 24x + 1 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 144x2 = (12x)2 Squaring last term 1 = (1)2 (a + b)2 = (12x)2 + 2 X 12x X 1 + (1)2 (a + b)2 = (12x + 1)2 E) 36y2 - 60y + 25 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 36y2 = (6y)2 Squaring last term 25 = (5)2 (a - b)2 = (6y)2 + 2 X 6y X 5 + (5)2 (a - b)2 = (6y - 5)2 F) 16x2 - 64 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 term 16x2 = (4x)2 Squaring term 64 = (8)2 a2 - b2 = (4x)2 - (8)2 a2 - b2 = (4x + 8) (4x - 8) G) x3 - x or x(x2 - 1) Now we can use formula a2 - b2 = (a + b) (a - b) and we get a2 - b2 = (x + 1) (x - 1) H) 4x2 - (2y - z)2 Squaring term 4x2 = (2x)2 and we get (2x)2 - (2y - z)2 Now we can apply the formula a2 - b2 = (a + b) (a - b) and we get a2 - b2 = [2x - (2y + z)] [2x - (2y - z)] Now will solve round brackets and we get a2 - b2 = [2x - 2y - z] [2x - 2y + z] Related Question Examples Factorize the following - A) 9y2 + 24y + 16 B) 9x2 - 16 C) 4y2 - 12y + 9 D) 144x2 + 24x + 1 E) 36y2 - 60y + 25 F) 16x2 - 64 G) x3 - x H) 4x2 - (2y - z)2