Factoring Polynomials Algebra I

Vocabulary Factors – The numbers used to find a product. Prime Number – A whole number greater than one and its only factors are 1 and itself. Composite Number – A whole number greater than one that has more than 2 factors.

Vocabulary Factored Form – A polynomial expressed as the product of prime numbers and variables. Prime Factoring – Finding the prime factors of a term. Greatest Common Factor (GCF) – The product of common prime factors.

Prime or Composite? Ex) 36 Ex) 23

Prime or Composite? Ex) 36 Composite. Factors: 1,2,3,4,6,9,12,18,36 Ex) 23 Prime. Factors: 1,23

Prime Factorization Ex) 90 = 2 ∙ 45 = 2∙ 3∙ 15 = 2∙ 3 ∙ 3 ∙ 5 OR use a factor tree:

Prime Factorization of Negative Integers Ex) -140 = -1 ∙ 140 = -1 ∙ 2 ∙ 70 = -1 ∙ 2 ∙ 7 ∙ 10 = -1 ∙ 2 ∙ 7 ∙ 2 ∙ 5

Now you try… Ex) 96 Ex) -24

Now you try… Ex) 96 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 Ex) ∙ 2 ∙ 2 ∙ 2 ∙ 3

Prime Factorization of a Monomial 12a²b³= 2 · 2 · 3 · a · a · b · b · b -66pq²= -1 · 2 · 3 · 11 · p · q · q

Finding GCF Ex) 48 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 60 = 2 ∙ 2 ∙ 3 ∙ 5 GCF = 2 · 2 · 3 = 12 Ex) 15 = 3 · 5 16 = 2 · 2 · 2 · 2 GCF – none = 1

Now you try… Ex) 36x²y 54xy²z

Now you try… Ex) 36x²y = 2 · 2 · 3 · 3 · x · x · y 54xy²z = 2 · 3 · 3 · 3 · x · y · y · z GCF = 18xy

Factoring Using the (Reverse) Distributive Property Factoring a polynomial means to find its completely factored form.

Factoring Using the (Reverse) Distributive Property First step is to find the prime factors of each term. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a

Factoring Using the (Reverse) Distributive Property First step is to find the prime factors of each term. Next step is to find the GCF of the terms in the polynomial. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a GCF = 4a

Factoring Using the (Reverse) Distributive Property First step is to find the prime factors of each term. Next step is to find the GCF of the terms in the polynomial. Now write what is left of each term and leave in parenthesis. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a 4a(3a + 4)

Factoring Using the (Reverse) Distributive Property First step is to find the prime factors of each term. Next step is to find the GCF of the terms in the polynomial. Now write what is left of each term and leave in parenthesis. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a 4a(3a + 4) Final Answer 4a(3a + 4)

Another Example: 18cd²+ 12c²d + 9cd

Another Example: 18cd²+ 12c²d + 9cd 18cd² = 2 · 3 · 3 · c · d · d 12c²d = 2 · 2 · 3 · c · c · d 9cd = 3 · 3 · c · d GCF = 3cd Answer: 3cd(6d + 4c + 3)