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Section P5 Factoring Polynomials

Section P5 Factoring Polynomials

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Section P5 Factoring Polynomials. Common Factors. Factoring a polynomial containing the sum of monomials mean finding an equivalent expression that is a product. In this section we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. - PowerPoint PPT Presentation

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Page 1: Section P5 Factoring Polynomials

Section P5Factoring Polynomials

Page 2: Section P5 Factoring Polynomials

Common Factors

Page 3: Section P5 Factoring Polynomials

Factoring a polynomial containing the sum of monomials mean finding an equivalent expression that is a product. In this section we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called prime.

Page 4: Section P5 Factoring Polynomials

Example

Factor:

2 3

2

64 28

5 ( 1) 10 ( 1)

x x

xy z xy z

Page 5: Section P5 Factoring Polynomials

Factoring by Grouping

Page 6: Section P5 Factoring Polynomials

Sometimes all of the terms of a polynomial may notcontain a common factor. However, by a suitable grouping of terms it may be possible to factor. Thisis called factoring by grouping.

Page 7: Section P5 Factoring Polynomials

Example

Factor by Grouping:

3 24 5 20x x x

Page 8: Section P5 Factoring Polynomials

Example

Factor by Grouping:

3 22 8 7 28x x x

Page 9: Section P5 Factoring Polynomials

Factoring Trinomials

Page 10: Section P5 Factoring Polynomials
Page 11: Section P5 Factoring Polynomials

Factors of 8

8,1 4,2 -8,-1 -4,-2

Sum of Factors

9 6 -9 -6

Factor:2 6 8x x

x x

4 2

+ +

Choose either two positive or two negative factors since the sign in front of the 8 is positive.

Page 12: Section P5 Factoring Polynomials

Factor: 22 9 5x x 2 x x +-

Possible factorizations Sum of outside and inside products

2 5 1

2 5 1

2 1 5

2 1 5

x x

x x

x x

x x

3

3

9

9

x

x

x

x

1 5

Since the sign in front of the 5 is a negative, one factor will be positive and one will be negative.

Page 13: Section P5 Factoring Polynomials

Example

Factor: 2 9 14x x

Possible Factorizations

Sum of Inside and Outside products

Page 14: Section P5 Factoring Polynomials

Example Factor: 23 2 21x x Possible Factorizations Sum of Inside and

outside Products

Page 15: Section P5 Factoring Polynomials

Factoring the Difference of

Two Squares

Page 16: Section P5 Factoring Polynomials
Page 17: Section P5 Factoring Polynomials
Page 18: Section P5 Factoring Polynomials

4

2 2

2

16

4 4

2 2 4

x

x x

x x x

Repeated Factorization- Another example

Can the sum of two squares be factored?

Page 19: Section P5 Factoring Polynomials

Example

Factor Completely:

24 9x

Page 20: Section P5 Factoring Polynomials

Example

Factor Completely:

249 81x

Page 21: Section P5 Factoring Polynomials

Example

Factor Completely:

4 481y x

Page 22: Section P5 Factoring Polynomials

Factoring Perfect Square Trinomials

Page 23: Section P5 Factoring Polynomials
Page 24: Section P5 Factoring Polynomials

Example

Factor:2 12 36x x

Page 25: Section P5 Factoring Polynomials

Example

Factor:

216 72 81x x

Page 26: Section P5 Factoring Polynomials

Factoring the Sum and Difference of Two Cubes

Page 27: Section P5 Factoring Polynomials
Page 28: Section P5 Factoring Polynomials

Example

Factor:

38 27x

Page 29: Section P5 Factoring Polynomials

Example

Factor:

3 3 3a b d

Page 30: Section P5 Factoring Polynomials

Example

Factor:

3 3125 64x y

Page 31: Section P5 Factoring Polynomials

A Strategy for Factoring Polynomials

Page 32: Section P5 Factoring Polynomials
Page 33: Section P5 Factoring Polynomials

Example

Factor Completely:3 212 60 75x x x

Page 34: Section P5 Factoring Polynomials

Example

Factor Completely:

4 81x

Page 35: Section P5 Factoring Polynomials

Example

Factor Completely:

327 64x

Page 36: Section P5 Factoring Polynomials

Example

Factor Completely:38 125x

Page 37: Section P5 Factoring Polynomials

Example

Factor Completely:

3 2 25 25x x x

Page 38: Section P5 Factoring Polynomials

Example

Factor Completely:

2 29 36x y

Page 39: Section P5 Factoring Polynomials

Factoring Algebraic Expressions Containing Fractional and Negative

Exponents

Page 40: Section P5 Factoring Polynomials

Expressions with fractional and negative exponents are not polynomials, but they can be factored using similar techniques. Find the greatest common factor with the smallest exponent in the terms.

3 1

4 4

31

4

3

4

3

4

3 6 6

6 3 6

6 4 6

2 6 2 3

x x x

x x x

x x

x x

3 11

4 4

Page 41: Section P5 Factoring Polynomials

Example

Factor and simplify:

1 3

4 4y y

Page 42: Section P5 Factoring Polynomials

Example

Factor and simplify:

1 3

2 25 5x x

Page 43: Section P5 Factoring Polynomials

Example

Factor and simplify:

2 1

3 33 3x x x

Page 44: Section P5 Factoring Polynomials

(a)

(b)

(c)

(d)

28 32x Factor Completely:

28 4

8 2 2

8 2 2

8 2 2

x

x x

x x

x x

Page 45: Section P5 Factoring Polynomials

(a)

(b)

(c)

(d)

Factor Completely:

3 327x y

2 2

2 2

2 2

2 2

(9 )(3 )

3

3 9 3

3 9 3

x y x y

x y x xy y

x y x xy y

x y x xy y