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Factoring Polynomials

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Factoring Polynomials

Algebra I

- 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.

- 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.

Ex) 36

Ex) 23

Ex) 36

Composite.

Factors: 1,2,3,4,6,9,12,18,36

Ex) 23

Prime.

Factors: 1,23

Ex) 90 = 2 ∙ 45

= 2∙ 3∙ 15

= 2∙ 3 ∙ 3 ∙ 5

OR use a factor tree:

90

9 10

3 3 2 5

Ex) -140 = -1 ∙ 140

= -1 ∙ 2 ∙ 70

= -1 ∙ 2 ∙ 7 ∙ 10

= -1 ∙ 2 ∙ 7 ∙ 2 ∙ 5

Ex) 96

Ex) -24

Ex) 96

2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3

Ex) -24

-1 ∙ 2 ∙ 2 ∙ 2 ∙ 3

12a²b³= 2 · 2 · 3 · a · a · b · b · b

-66pq²= -1 · 2 · 3 · 11 · p · q · q

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

Ex) 36x²y

54xy²z

Ex) 36x²y = 2 · 2 · 3 · 3 · x · x · y

54xy²z = 2 · 3 · 3 · 3 · x · y · y · z

GCF = 18xy

- Factoring a polynomial means to find its completely factored form.

- 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

- 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

- 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)

- 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)

18cd²+ 12c²d + 9cd

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)