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TMAT 103
Chapter 5
Factoring and Algebraic Fractions
TMAT 103
§5.1
Special Products
§5.1 – Special Products
• a(x + y + z) = ax + ay + az
• (x + y)(x – y) = x2 – y2
• (x + y)2 = x2 + 2xy +y2
• (x – y)2 = x2 – 2xy +y2
• (x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz
• (x + y)3 = x3 + 3x2y + 3xy2 + y3
• (x – y)3 = x3 – 3x2y + 3xy2 – y3
TMAT 103
§5.2
Factoring Algebraic Expressions
§5.2 – Factoring Algebraic
Expressions
• Greatest Common Factor
ax + ay + az = a(x + y + z)
• Examples – Factor the following
3x – 12y
40z2 + 4zx – 8z3y
§5.2 – Factoring Algebraic
Expressions
• Difference of two perfect squares
x2 – y2 = (x + y)(x – y)
• Examples – Factor the following
16a2 – b2
36a2b4 – 100a4z10
256x4 – y16
§5.2 – Factoring Algebraic
Expressions
• General trinomials with quadratic coefficient 1
x2 + bx + c
• Examples – Factor the following
x2 + 8x + 15
q2 – 3q – 28
x2 + 3x – 4
2m2 – 18m + 28
b4 + 21b2 – 100
x2 + 3x + 1
§5.2 – Factoring Algebraic
Expressions
• Sign Patterns
Equation
Template
x2 + bx + c
( + )( + )
x2 + bx – c
( + )( – )
x2 – bx + c
( – )( – )
x2 – bx – c
( + )( – )
§5.2 – Factoring Algebraic
Expressions
• General trinomials with quadratic coefficient
other than 1
ax2 + bx + c
• Examples – Factor the following
6m2 – 13m + 5
9x2 + 42x + 49
9c4 – 12c2y2 + 4y4
TMAT 103
§5.3
Other Forms of Factoring
§5.3 – Other Forms of Factoring
• Examples – Factor the following
a(b + m) – c(b + m)
4x + 2y + 2cx + cy
x3 – 2x2 + x – 2
36q2 – (3x – y)2
y2 + 6y + 9 – 49z4
(m – n)2 – 6(m – n) + 9
§5.3 – Other Forms of Factoring
• Sum of two perfect cubes
x3 + y3 = (x + y)(x2 – xy + y2)
• Examples – Factor the following
x3 + 64
8z3m6 + 27p9
§5.3 – Other Forms of Factoring
• Difference of two perfect cubes
x3 – y3 = (x – y)(x2 + xy + y2)
• Examples – Factor the following
m3 – 125
8z3 – 64p9s3
TMAT 103
§5.4
Equivalent Fractions
§5.4 – Equivalent Fractions
• A fraction is in lowest terms when its
numerator and denominator have no
common factors except 1
• The following are equivalent fractions
a
b
=
ax
bx
§5.4 – Equivalent Fractions
• Examples – Reduce the following fractions to lowest
terms
x2 – 2x – 24
2x2 + 7x – 4
a2 – ab + 3a – 3b
a2 – ab
x4 – 16
x4 – 2x2 – 8
x3 – y3
x2 – y2
TMAT 103
§5.5
Multiplication and Division of
Algebraic Fractions
§5.5 – Multiplication and
Division of Algebraic Fractions
• Multiplying fractions
a
b
•
c
d
=
ac
bd
.
• Dividing fractions
a  c
b
d
=
a
b
•
d
c
=
ad
bc
.
§5.5 – Multiplication and
Division of Algebraic Fractions
• Examples – Perform the indicated operations and
simplify
4t4 • 12t2
6t
9t3
a2 – a – 2
a2 + 7a + 6
•
a2 + 3a – 18
a2 – 4a + 4
4
15pq2
39mn

13m5n3
5p4q3
TMAT 103
§5.6
Addition and Subtraction of
Algebraic Fractions
§5.6 Addition and Subtraction of
Algebraic Fractions
•
Finding the lowest common denominator (LCD)
1. Factor each denominator into its prime factors; that is,
factor each denominator completely
2. Then the LCD is the product formed by using each of
the different factors the greatest number of times that
it occurs in any one of the given denominators
§5.6 Addition and Subtraction of
Algebraic Fractions
• Examples – Find the LCD for:
2
8
, 125 , and 307
4
x
, y32 , and xy52
4
x 2  6 x 9
,
3
( x 3)
, and
5
x 2 9
§5.6 Addition and Subtraction of
Algebraic Fractions
•
Adding or subtracting fractions
1. Write each fraction as an equivalent fraction over the
LCD
2. Add or subtract the numerators in the order they
occur, and place this result over the LCD
3. Reduce the resulting fraction to lowest terms
§5.6 Addition and Subtraction of
Algebraic Fractions
• Perform the indicated operations
4
s 3
 1s
1
6x

2
x2  y2
1
3 x 6


1
2 x4
1
x 2 3 xy  2 y 2

3
x 2  xy  2 y 2
TMAT 103
§5.7
Complex Fractions
§5.7 Complex Fractions
•
A complex fraction that contains a fraction in the
numerator, denominator, or both. There are 2
methods to simplify a complex fraction
–
Method 1
•
–
Multiply the numerator and denominator of the complex
fraction by the LCD of all fractions appearing in the
numerator and denominator
Method 2
•
Simplify the numerator and denominator separately. Then
divide the numerator by the denominator and simplify
again.
§5.7 Complex Fractions
• Use both methods to simplify each of the
complex fractions
2 1
c
2 1
c
3
5x
x2 4
3 x 2 2
TMAT 103
§5.8
Equations with Fractions
§5.8 Equations with Fractions
•
To solve an equation with fractions:
1. Multiply both sides by the LCD
2. Check
•
Equations MUST BE CHECKED for
extraneous solutions
–
–
Multiplying both sides by a variable may
introduce extra solutions
Consider x = 3, multiply both sides by x
§5.8 Equations with Fractions
• Solve and check
4 x 3
9
2 x 5
x
2 
x 4
3
 
2
x
Solve V 
Q
R1
1
4

Q
R2
for R 2