1. Chapter 5 Factoring and Algebraic Fractions
TMAT 103
Chapter 5
Factoring and Algebraic Fractions

2. TMAT 103
§5.1
Special Products

3. §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

4. §5.2 Factoring Algebraic Expressions
TMAT 103
§5.2
Factoring Algebraic Expressions

5. §5.2 – Factoring Algebraic Expressions
Greatest Common Factor
ax + ay + az = a(x + y + z)
Examples – Factor the following
3x – 12y
40z2 + 4zx – 8z3y

6. §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

7. §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

8. §5.2 – Factoring Algebraic Expressions
Sign Patterns
Equation
Template
x2 + bx + c
( + )( + )
x2 + bx – c
( + )( – )
x2 – bx + c
( – )( – )
x2 – bx – c

9. §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

10. §5.3 Other Forms of Factoring
TMAT 103
§5.3
Other Forms of Factoring

11. §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

12. §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

13. §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

14. §5.4 Equivalent Fractions
TMAT 103
§5.4
Equivalent Fractions

15. §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 = ax b bx

16. §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

17. §5.5 Multiplication and Division of Algebraic Fractions
TMAT 103
§5.5
Multiplication and Division of Algebraic Fractions

18. §5.5 – Multiplication and Division of Algebraic Fractions
Multiplying fractions
a • c = ac
b d bd
Dividing fractions
a  c = a • d = ad
b d b c bc

19. §5.5 – Multiplication and Division of Algebraic Fractions
Examples – Perform the indicated operations and simplify
4t4 • 12t2
6t t3
a2 – a – • a2 + 3a – 18
a2 + 7a a2 – 4a + 4
15pq  39mn4
13m5n p4q3

20. §5.6 Addition and Subtraction of Algebraic Fractions
TMAT 103
§5.6
Addition and Subtraction of Algebraic Fractions

21. §5.6 Addition and Subtraction of Algebraic Fractions
Finding the lowest common denominator (LCD)
Factor each denominator into its prime factors; that is, factor each denominator completely
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

22. §5.6 Addition and Subtraction of Algebraic Fractions
Examples – Find the LCD for:

23. §5.6 Addition and Subtraction of Algebraic Fractions
Adding or subtracting fractions
Write each fraction as an equivalent fraction over the LCD
Add or subtract the numerators in the order they occur, and place this result over the LCD
Reduce the resulting fraction to lowest terms

24. §5.6 Addition and Subtraction of Algebraic Fractions
Perform the indicated operations

25. TMAT 103
§5.7
Complex Fractions

26. §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.

27. §5.7 Complex Fractions
Use both methods to simplify each of the complex fractions

28. §5.8 Equations with Fractions
TMAT 103
§5.8
Equations with Fractions

29. §5.8 Equations with Fractions
To solve an equation with fractions:
Multiply both sides by the LCD
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

30. §5.8 Equations with Fractions
Solve and check