Download R.5 - Gordon State College

Sullivan Algebra and
Trigonometry: Section R.5
Factoring
Objectives of this Section
• Factor the Difference of Two Squares
• Factor the Sum and Difference of Two Cubes
• Factor Second Degree Polynomials
• Factor By Grouping
The process of expressing a polynomial
as a product of other polynomials is
called factoring.
Example
Multiply: 3 x (x - 2 x - 4)
2
= 3 x ( x 2 ) - 3 x (2 x) - 3 x(4)
= 3 x 3 - 6 x 2 - 12 x
Factoring is the same process in reverse
3
2
Factor: 3 x - 6 x - 12 x
Notice that each term in this trinomial has
a greatest common factor of 3x.
3
2
3 x - 6 x - 12 x
= 3 x( x 2 ) - 3 x(2 x) - 3 x(4)
= 3 x (x 2 - 2 x - 4)
Special Formulas
When you factor a polynomial, first check
whether you can use one of the special
formulas shown in the previous section.
Difference of Two Squares: x 2 - a 2 = ( x - a )( x + a )
Perfect Squares:
2
x 2 + 2ax + a 2 = (x + a )
2
x 2 - 2ax + a 2 = (x - a )
Sum of Two Cubes: x3 + a 3 = (x + a )(x 2 - ax + a 2 )
3
3
2
2
x
a
=
x
a
x
+
ax
+
a
(
)
(
)
Difference of Two Cubes:
Example:
Factor Completely: 9 x 2 - 64
9 x  64   3 x   8
2
2
2
  3 x  8 3 x  8
Factor Completely: x 4  1
x  1   x  1  x  1
4
2

x
2
2
1
 x  1 x  1
Factor the trinomial: x 2  11x  18
Look for factors of 18 whose sum is 11.
9  2  18
9  2  11
x  11x  18   x  2  x  9 
2
Factor the trinomial: x  3x  10
2
Factors of -10
10, -1
Sum
9
5, -2
3
-5, 2
-3
-10, 1
-9
x  3 x  10   x  5 x  2 
2
Factor completely: 3x  7 x  6
2
3x  7 x  6   3x
2
 3 x
 3 x

2
3x  7 x  6  
3
x


 3x
 x

1 x 6
6 x 1
2 x 3
3 x
2
 3x + 1 x  6
 3x  1 x  6

 3x  6 x  1

 3x  6 x  1
2
3x  7 x  6  
 3x  2 x  3
 3x  2 x  3

 3x  3 x  2
 3x  3 x  2
Factor By Grouping:
2 x  10x  3x  15
3
2


 2 x  10 x  3x  15
3
2
 2 x  x  5  3 x  5
2


  x  5 2 x  3
2