Section P5 Factoring Polynomials. Common Factors.
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Transcript of Section P5 Factoring Polynomials. Common Factors.
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Section P5Factoring Polynomials
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Common Factors
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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.
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Example
Factor:
2 3
2
64 28
5 ( 1) 10 ( 1)
x x
xy z xy z
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Factoring by Grouping
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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.
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Example
Factor by Grouping:
3 24 5 20x x x
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Example
Factor by Grouping:
3 22 8 7 28x x x
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Factoring Trinomials
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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.
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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.
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Example
Factor: 2 9 14x x
Possible Factorizations
Sum of Inside and Outside products
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Example Factor: 23 2 21x x Possible Factorizations Sum of Inside and
outside Products
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Factoring the Difference of
Two Squares
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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?
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Example
Factor Completely:
24 9x
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Example
Factor Completely:
249 81x
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Example
Factor Completely:
4 481y x
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Factoring Perfect Square Trinomials
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Example
Factor:2 12 36x x
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Example
Factor:
216 72 81x x
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Factoring the Sum and Difference of Two Cubes
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Example
Factor:
38 27x
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Example
Factor:
3 3 3a b d
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Example
Factor:
3 3125 64x y
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A Strategy for Factoring Polynomials
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Example
Factor Completely:3 212 60 75x x x
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Example
Factor Completely:
4 81x
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Example
Factor Completely:
327 64x
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Example
Factor Completely:38 125x
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Example
Factor Completely:
3 2 25 25x x x
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Example
Factor Completely:
2 29 36x y
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Factoring Algebraic Expressions Containing Fractional and Negative
Exponents
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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
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Example
Factor and simplify:
1 3
4 4y y
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Example
Factor and simplify:
1 3
2 25 5x x
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Example
Factor and simplify:
2 1
3 33 3x x x
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(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
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(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