Transcript Algebra Expressions and Real Numbers

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.
Polynomials that cannot be factored using integer
coefficients are called prime.
Example
Factor:
64 x  28 x
2
3
5 xy ( z  1)  10 xy ( z  1)
2
Factoring by Grouping
Sometimes all of the terms of a polynomial may not
contain a common factor. However, by a suitable
grouping of terms it may be possible to factor. This
is called factoring by grouping.
Example
Factor by Grouping:
x  4 x  5 x  20
3
2
Example
Factor by Grouping:
2 x  8x  7 x  28
3
2
Factoring Trinomials
Factor:
x  6x  8
2
 x +  x + 
Factors
of 8
Sum of
Factors
8,1 4,2
4 2 -8,-1 -4,-2
9
6
-9
-6
Choose either two positive or two negative factors
since the sign in front of the 8 is positive.
Factor:
2x  9x  5
2
 2x -  x + 
Possible factorizations
 2 x  5 x  1
 2 x  5 x  1
 2 x 11 x  55
 2 x  1 x  5
Sum of outside and inside
products
3x
3 x
9x
9 x
Since the sign in front of the 5 is a negative, one
factor will be positive and one will be negative.
Example
Factor:
x  9 x  14
2
Possible
Factorizations
Sum of Inside
and Outside
products
Example
Factor:
3x  2 x  21
2
Possible Factorizations Sum of Inside and
outside Products
Factoring the Difference of
Two Squares
Repeated Factorization- Another example
x  16
4
x
2
 4  x  4 
2
 x  2  x  2   x
2
 4
Can the sum of two squares be factored?
Example
Factor Completely:
4x  9
2
Example
Factor Completely:
49 x  81
2
Example
Factor Completely:
y  81x
4
4
Factoring Perfect Square
Trinomials
Example
Factor:
x  12 x  36
2
Example
Factor:
16 x  72 x  81
2
Factoring the Sum and
Difference of Two Cubes
Example
Factor:
8 x  27
3
Example
Factor:
a b d
3 3
3
Example
Factor:
125x  64 y
3
3
A Strategy for Factoring
Polynomials
Example
Factor Completely:
12 x 3  60 x 2  75 x
Example
Factor Completely:
x  81
4
Example
Factor Completely:
27 x  64
3
Example
Factor Completely:
8 x  125
3
Example
Factor Completely:
x  x  25 x  25
3
2
Example
Factor Completely:
9 x  36 y
2
2
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
4
3x  x  6    x  6 

 x  6

3
4
 3x   x  6  
1
3
4
 x  6  4x  6

2  x  6
1
4

3
4
 2 x  3
3
1
 1 
4
4
Example
Factor and simplify:
1
4
y y
3
4
Example
Factor and simplify:
 x  5

1
2
  x  5

3
2
Example
Factor and simplify:
x  x  3
2

3
  x  3
1
3
Factor Completely:
8 x  32
2
2
8
x
(a)   4 


(b) 8 x  2  x  2 
(c)  8 x  2  x  2 
(d) 8  x  2  x  2 
Factor Completely:
27x  y
3
3
(a) (9 x 2  y )(3x  y 2 )
(b)  x  y   x 2  3xy  y 2 
(c)  3x  y   9 x 2  3xy  y 2 
(d)  3x  y   9 x 2  3xy  y 2 