Factoring polynomials
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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.

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Factoring Polynomials

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Factoring polynomials

Factoring Polynomials

Algebra I


Vocabulary

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.


Vocabulary1

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

Prime or Composite?

Ex) 36

Ex) 23


Prime or composite1

Prime or Composite?

Ex) 36

Composite.

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

Ex) 23

Prime.

Factors: 1,23


Prime factorization

Prime Factorization

Ex) 90 = 2 ∙ 45

= 2∙ 3∙ 15

= 2∙ 3 ∙ 3 ∙ 5

OR use a factor tree:

90

9 10

3 3 2 5


Prime factorization of negative integers

Prime Factorization of Negative Integers

Ex) -140 = -1 ∙ 140

= -1 ∙ 2 ∙ 70

= -1 ∙ 2 ∙ 7 ∙ 10

= -1 ∙ 2 ∙ 7 ∙ 2 ∙ 5


Now you try

Now you try…

Ex) 96

Ex) -24


Now you try1

Now you try…

Ex) 96

2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3

Ex) -24

-1 ∙ 2 ∙ 2 ∙ 2 ∙ 3


Prime factorization of a monomial

Prime Factorization of a Monomial

12a²b³= 2 · 2 · 3 · a · a · b · b · b

-66pq²= -1 · 2 · 3 · 11 · p · q · q


Finding gcf

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 try2

Now you try…

Ex) 36x²y

54xy²z


Now you try3

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 Using the (Reverse) Distributive Property

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


Factoring using the reverse distributive property1

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 property2

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 property3

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 property4

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

Another Example:

18cd²+ 12c²d + 9cd


Another example1

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)


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