section p5 factoring polynomials. common factors
TRANSCRIPT
Section P5Factoring 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. Polynomials that cannot be factored using integer coefficients are called prime.
Example
Factor:
2 3
2
64 28
5 ( 1) 10 ( 1)
x x
xy z xy z
Factoring by Grouping
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.
Example
Factor by Grouping:
3 24 5 20x x x
Example
Factor by Grouping:
3 22 8 7 28x x x
Factoring Trinomials
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.
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.
Example
Factor: 2 9 14x x
Possible Factorizations
Sum of Inside and Outside products
Example Factor: 23 2 21x x Possible Factorizations Sum of Inside and
outside Products
Factoring the Difference of
Two Squares
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?
Example
Factor Completely:
24 9x
Example
Factor Completely:
249 81x
Example
Factor Completely:
4 481y x
Factoring Perfect Square Trinomials
Example
Factor:2 12 36x x
Example
Factor:
216 72 81x x
Factoring the Sum and Difference of Two Cubes
Example
Factor:
38 27x
Example
Factor:
3 3 3a b d
Example
Factor:
3 3125 64x y
A Strategy for Factoring Polynomials
Example
Factor Completely:3 212 60 75x x x
Example
Factor Completely:
4 81x
Example
Factor Completely:
327 64x
Example
Factor Completely:38 125x
Example
Factor Completely:
3 2 25 25x x x
Example
Factor Completely:
2 29 36x y
Factoring Algebraic Expressions Containing Fractional and Negative
Exponents
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
Example
Factor and simplify:
1 3
4 4y y
Example
Factor and simplify:
1 3
2 25 5x x
Example
Factor and simplify:
2 1
3 33 3x x x
(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
(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