Download Factoring (pp. 1 Of 4)

Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 1 of 4)
Try these items from middle school math.
Review
A) What numbers are the
factors of 24?

B) Write down the prime
factorization of 72.
C) Simplify 36
using the
48
greatest common factor
(CGF).
What does it mean to “factor” a number (or, to “find its factors”)?
(2 x  3 y )( 4 x  y )  8 x 2  2 xy  12 xy  3 y 2  8 x 2  14 xy  3 y 2
Polynomials:

Consider the polynomial problem above. What are the “factors”?

Look back at the previous activity (“Fact”-ors About Islands). What are the factors of x 2  49 ?

What does it mean to “factor” a polynomial (or, to “find its factors”)?
Factoring Polynomials
Step One:
Step Two:
Follow these steps to factor polynomials.
Always look to see if the terms have a greatest common factor (GCF) (other than 1)
Example: 6 x 2  14 x
Example: 4a 2 b  6ab 2  10ab
GCF =
GCF =
Factors: _____ ( _____ + _____ )
Factors: _____ ( _____ + _____ + _____ )
After checking for the GCF, remaining polynomials can be factored by several different
methods, according to the number of terms in the polynomial.
A) Two Terms
1. Difference of Squares: a 2  b 2  (a  b )(a  b )
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Example: 9 x 2  64 y 2
Example: 20 x 2  45
GCF =
GCF =
Factors:
Factors:
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 2 of 4)
Two Terms (continued)
2. Difference of Cubes: a 3  b 3  (a  b )(a 2  ab  b 2 )
Example: x 3  y 3
GCF =
Factors:
3. Sum of Cubes: a 3  b 3  (a  b )(a 2  ab  b 2 )
Example: 16m 3  2 p 3
GCF =
Factors:
B) Three Terms
1. Leading Coefficient of 1
Example: x 2  4 x  12
Example: 2 x 2  10 x  12
GCF =
GCF =
Factors:
Factors:
2. Leading Coefficient other than 1
Various Methods
Example: 6 x 2  7 x  5
GCF =
Factors:
©2010, TESCCC
08/01/10

Guess and check

Box method
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 3 of 4)
©2010, TESCCC
08/01/10

Gross Product

Bottoms up
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 4 of 4)
C) Four Terms
 Grouping
Example: 12xy + 2x + 30y + 5
Example: 6ab + 2a + 15b + 5
___ ( ___ + ___ ) + ___ ( ___ + ___ )
( ___ + ___ )( ___ + ___ )
Practice Problems
Factor the following polynomials.
1)
14 x 2 y  4 xy 2 2 xy
2)
4x 2  9
3)
2a 3  2ab 2
4)
x 3  27
5)
64 x 3  1
6)
x3y + 8y
7)
x 2  14 x  49
8)
2 x 2  6 x  56
9)
y 2  3 y  54
10)
4 y 2  11y  3
11)
5a 2  22a  8
12)
36 x 2  6 x  20
13)
6 x 2  9 x  81
14)
ab  9a  9b  81
15)
4 xy  8 x  7 y  14
16) The area of a right triangle is represented by the expression 6x2 + 5x – 4. If the height of the
triangle is represented by the expression 3x + 4, find an expression to represent the base.
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 1 of 4) KEY
Try these items from middle school math.
Review
A) What numbers are the
factors of 24?
B) Write down the prime
factorization of 72.
22233, or 2332
1, 2, 3, 4, 6, 8, 12, 24
C) Simplify 36
using the
48
greatest common factor
(CGF).
3
123
=
4
124
 What does it mean to “factor” a number (or, to “find its factors”)?
Answers will vary. Sample: Find numbers that multiply to give you the number.
(2 x  3 y )( 4 x  y )  8 x 2  2 xy  12 xy  3 y 2  8 x 2  14 xy  3 y 2
Polynomials:

Consider the polynomial problem above. What are the “factors”? (2x + 3y) and (4x + y)

Look back at the previous activity (“Fact”-ors About Islands). What are the factors of x 2  49 ?
(x + 7)(x – 7)
 What does it mean to “factor” a polynomial (or, to “find its factors”)?
Answers will vary. Sample: Find polynomials you could multiply to give you a certain answer.
Factoring Polynomials
Step One:
Step Two:
Follow these steps to factor polynomials.
Always look to see if the terms have a greatest common factor (GCF) (other than 1)
Example: 6 x 2  14 x
Example: 4a 2 b  6ab 2  10ab
GCF = 2x
GCF = 2ab
Factors: 2x (3x + 7)
Factors: 2ab (2a + 3b + -5)
After checking for the GCF, remaining polynomials can be factored by several different
methods, according to the number of terms in the polynomial.
A) Two Terms
1. Difference of Squares: a 2  b 2  (a  b )(a  b )
©2010, TESCCC
Example: 9 x 2  64 y 2
Example: 20 x 2  45
GCF = None (other than 1)
GCF = 5
Factors: (3x + 8y)(3x – 8y)
Factors: 5(4x2 – 9) = 5(2x + 3)(2x – 3)
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 2 of 4) KEY
Two Terms (continued)
2. Difference of Cubes: a 3  b 3  (a  b )(a 2  ab  b 2 )
Example: x 3  y 3
GCF = None (other than 1)
 Square of 1st term
 Terms multiplied and
take the opposite sign
 Square of 2nd term
2
Factors: (x – y)(x + xy + y)
3. Sum of Cubes: a 3  b 3  (a  b )(a 2  ab  b 2 )
Example: 16m 3  2 p 3
GCF = 2
Factors: 2(8m3 + p3) = 2(2m + p)(4m2 – 2mp + p2)
B) Three Terms
1. Leading Coefficient of 1
Example: x 2  4 x  12
Example: 2 x 2  10 x  12
GCF = None (other than 1)
GCF = 2
Factors: (x + 6)(x – 2)
Factors: 2(x2 + 5x – 6) = 2(x + 6)(x – 1)
Find factors of the last term (-12), that
combine to give the middle term (+4)
(3)(-4)NO
(-3)(4)NO
(6)(-2)YES (-6)(2)NO
After GCF, Find factors of the last term
(-6), that combine to give the middle term
(+5)
(-2)(3)NO
(2)(-3)NO
(6)(-1)YES (-6)(1)NO
2. Leading Coefficient other than 1
Various Methods
Example: 6 x 2  7 x  5
GCF = None (other than 1)
Factors: (3x + 5)(2x – 1)
©2010, TESCCC
08/01/10

Guess and check

Box method
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 3 of 4) KEY
 Gross Product
(Illustrated below)
Find the product of the leading
coefficient and constant.
6  5 = 30
Determine two factors of this product that
combine to give the middle term.
Factors of 30 that combine to +7
+10 and -3
Replace the middle term with two x
terms with these coefficients.
6 x 2  10 x  3 x  5
Group as binomials. (See four terms.)
 6x 2  10x    3 x  5 
Factor each binomial.
2 x  3 x  5   1 3 x  5 
Factor out the common factor and group
the remaining terms.
 3 x  5  2 x  1
 Bottoms up
(Illustrated below)
Multiply leading coefficient and constant
and put result in as the final term.
x2 + 7x – 30
Factor as before as you would with a
leading coefficient of 1.
(x + 10)(x – 3)
Divide the 6 out of the last terms.
(x + 10/6)(x – 3/6)
Simplify rational numbers.
(x + 5/3)(x – ½)
“Bottoms up” the remaining
denominators.
(3x +5)(2x -1)
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Factoring (pp. 4 of 4) KEY
C) Four Terms
 Grouping
Example: 12xy + 2x + 30y + 5
Example: 6ab + 2a + 15b + 5
2x ( 6y + 1) + 5 (6y + 1 )
2a(3b + 1) + 5(3b + 1)
(6y + 1)(2x + 5)
(3b + 1)(2a + 5)
Practice Problems
Factor the following polynomials.
1)
14 x 2 y  4 xy 2 2 xy
2)
2xy(7x + 2y + 1)
3)
2a 3  2ab 2
(2y – 3)(2y + 3)
4)
64 x 3  1
6)
(4x – 1)(16x2 + 4x + 1)
7)
x 2  14 x  49
y 2  3 y  54
8)
5a 2  22a  8
10)
6 x 2  9 x  81
12)
36 x 2  6 x  20
2(6x + 5)(3x – 2)
14)
3(2x – 9)(x + 3)
15)
4 y 2  11y  3
(y + 3)(4y – 1)
(a – 4)(5a – 2)
13)
2 x 2  6 x  56
2(x + 7)(x – 4)
(y – 9)(y + 6)
11)
x3y + 8y
y(x + 2)(x2 – 2x + 4)
(x – 7)(x – 7) or (x – 7)2
9)
x 3  27
(x + 3)(x2 – 3x + 9)
2a(a – b)(a + b)
5)
4x 2  9
ab  9a  9b  81
(b – 9)(a + 9)
4 xy  8 x  7 y  14
(y – 2)(4x + 7)
16) The area of a right triangle is represented by the expression 6x2 + 5x – 4. If the height of the
triangle is represented by the expression 3x + 4, find an expression to represent the base.
6x2 + 5x – 4 = ½ (3x + 4)b
2(6x2 + 5x – 4) = (3x + 4)b
2(3x +4)(2x – 1) =(3x + 4)b
b = 2(2x – 1) or 4x – 2
©2010, TESCCC
08/01/10