Factoring Polynomials Part 1 GREATEST COMMON FACTOR GCF

Factoring Polynomials: Part 1 GREATEST COMMON FACTOR (GCF) is the product of all prime factors that are shared by all terms and the smallest exponent of any variable common to all terms. • LARGEST NUMBER that can divide all terms • SMALLEST EXPONENT of common variables to all terms Examples: 1) 15, 16 2) 72, 36, 42 3) 6 x 2 y 6, 32 x 3 y 4, 10 x 5 y 3 72 = 2 • 2 • 3 6 x 2 y 6 = 2 • 3 • x 2 • y 6 15 = 3 • 5 16 = 2 • 2 • 2 36 = 2 • 3 • 3 32 x 3 y 4 = 2 • 2 • x 3 • y 4 42 = 2 • 3 • 7 10 x 5 y 3= 2 • 5 • x 5 • y 3 GCF = 2 • 3 = 6 GCF = 2 • x 2 • y 3 = 2 x 2 y 3 GCF = 1

Find the GREATEST COMMON FACTOR (GCF) for each of the following group of terms (1) 45, 75 (2) 14, 49 (3) 36, 90, 42 (4) 36 x 2 y, 54 xy 2 z (5) 45 a 5 b 3, 60 a 2 b 7 (6) 4 pq 3, 15 qr 2 (7) 27 xy, 18 x, 12 x 2 (8) 144 a 2 b 2, 36 ab 2, (9) 25 r 3 s 6 t 2, 12 ab 15 r 5 s 2 t 3, 75 r 4 s 3 t

Factoring a polynomial by GREATEST COMMON FACTOR (GCF) “Reverse the Distributive Property” STEP #1: Find the GCF for all terms of polynomial STEP #2: Find Factored Polynomial by dividing all terms by GCF STEP #3: Factored Form = (Step #1)(Step #2) Exp 1: 10 x 3 z – 25 x 6 y Exp 2: 14 a 3 b 2 + 28 a 5 b 5 + 35 a 2 b 4 Step #1: GCF = 5 x 3 Step #1: Step #2: Step #3: 5 x 3(2 z – 5 x 3 y) Step #3:

FACTORING PRACTICE #1: Factor by the GCF (1) 72 a 3 – 50 ab 2 (2) 6 y 5 + 30 y 4 + 24 y 3 (3) 10 x 2 – 45 x + 35 (4) 2 xy – 10 x (5) 5 x 2 y 2 – 15 x 2 y (6) 12 x 2 – 42 xy + 9 y 2

Factoring Polynomials: Part 2 Special Binomial (2 Term) Factoring Techniques [1] Difference of Squares MEMORIZE!! EXP#1: EXP#2: EXP#3:

FACTORING PRACTICE #2: Difference of Squares a) b) c) d) e) f)

[2] Sum and Difference of Cubes MEMORIZE! EXP#1: MEMORIZE! EXP#2:

FACTORING PRACTICE #3: Sum and Difference of Cubes a) b) c) d) e) f)

FACTORING PRACTICE #4: Check for GCF, then 2 Term Cases a) b) c) d) e) f)

Factoring Polynomials: Part 3 4 – Term Polynomials STEP #1: Check for GCF of entire polynomial STEP #2: Factor by Grouping • Split polynomial: FIRST two terms and the LAST two terms. • FACTOR the GCF from both sides of split • Check for negative and positive sign agreement • Factored Form: (1 st GCF + 2 nd GCF) (factored polynomial) Algebraic Example: 10 x 2 + 5 x + 6 x + 3 GCF = 5 x GCF = 3 5 x(2 x + 1) + 3(2 x + 1) a is first GCF and d is second GCF (5 x+ 3) (2 x + 1)

FACTORING PRACTICE #5: Factoring by Grouping a) c) b) d)

e) g) f) h)