In mathematics, factorization or factoring is the decomposition of an object (for example, a number or a polynomial) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5 In all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to "basic building blocks”

ftrees

To factor an expression means to: re-write it as a product Why factor? Once an expression has been factored, equivalent factors can divide to one

Remember factor trees? 9 33X 91X 9 = 3 X 3

Notice: 4(x + 2) = 4x + 8 Expanding Factoring

Common Factoring A common factor can be divided out of every term in the polynomial 1.Number: GCF 2.Variable: Common Variable, take the lowest exponent

Common Factor: 10x + 20 =10( ) 25x x =25x( ) 4x + 12x 2 – 16x 3 = 4x(1 + 3x – 4x 2 ) 8a 2 b a 4 b 2 = 4a 2 b 2 (2b + 3a 2 ) 10 x+ 2 25x x + 4

YOU MAY DIVIDE OUT COMMON FACTORS YOU MAY NOT “CROSS OUT” OR “CANCEL” SIMILAR TERMS Consider the following example:

3x + 6 3, x = 2 Direct = 3(2) = 12 3 = 4 Factoring 3x = 3(x + 2) = x + 2 = 4

Crossing out 3x = x + 6 = 8

Stop here…

Trinomials: ax 2 + bx + c Simple (a = 1): Add to the middle Multiply to the last

Factor x 2 + 5x + 6 =(x )(x ) simple Add: 5 Multiply: 6

Factor x 2 - 2x - 15 =(x )(x ) simple Add: -2 Multiply: -15

Complex: (a > 1) Decomposition Add to the middle, multiply to (first)(last) Common Factor twice

Factor 6x 2 – 1x – 2 =6x 2 – 4x + 3x – 2 = 2x(3x – 2) + 1(3x – 2) = (3x – 2)(2x + 1) You may also guess and check complex Add: -1 Multiply: -12 CF the first pair, then CF the second pair

Difference of Squares x 2 – y 2 =(x – y)(x + y) x 2 – 25 =(x – 5)(x + 5) 4x 2 – 36y 2 =(2x – 6y)(2x + 6y)

Grouping Sometimes, part of a 4 term polynomial can be “grouped” together and factored

Factor ax + cx + ay + cy = x(a + c) + y(a + c) = (a + c)(x + y)

Factor a 2 – p 2 + 2a + 1 = a 2 + 2a + 1 – p 2 =(a + 1) 2 – p 2 =(a p)(a p)

The Great Factoring Plan!!! C.F D of S Simple Complex Grouping

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