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 Factorax + ay + az = a(x + y + z)

• Examples – Factor the following3x – 12y

40z2 + 4zx – 8z3y

§5.2 – Factoring Algebraic Expressions

• Difference of two perfect squaresx2 – y2 = (x + y)(x – y)

• Examples – Factor the following16a2 – b2

36a2b4 – 100a4z10

256x4 – y16

§5.2 – Factoring Algebraic Expressions

• General trinomials with quadratic coefficient 1x2 + bx + c

• Examples – Factor the followingx2 + 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 1ax2 + bx + c

• Examples – Factor the following6m2 – 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 followinga(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 cubesx3 + y3 = (x + y)(x2 – xy + y2)

• Examples – Factor the followingx3 + 64

8z3m6 + 27p9

§5.3 – Other Forms of Factoring

• Difference of two perfect cubesx3 – y3 = (x – y)(x2 + xy + y2)

• Examples – Factor the followingm3 – 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 = ax

b bx

§5.4 – Equivalent Fractions

• Examples – Reduce the following fractions to lowest termsx2 – 2x – 242x2 + 7x – 4

a2 – ab + 3a – 3b a2 – ab

x4 – 16x4 – 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 • c = ac .

b d bd

• Dividing fractions a c = a • d = ad .

b d b c bc

§5.5 – Multiplication and Division of Algebraic Fractions

• Examples – Perform the indicated operations and simplify4t4 • 12t2

6t 9t3

a2 – a – 2 • a2 + 3a – 18

a2 + 7a + 6 a2 – 4a + 4

15pq2 39mn4

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:

307

125

82 ,, and

22534 and ,,xyyx

95

)3(3

964

22 and ,, xxxx

§5.6 Addition and Subtraction of Algebraic Fractions

• Adding or subtracting fractions1. 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

ss1

34

421

631

61

xxx

222222 23

2312

yxyxyxyxyx

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

1

12

2

c

c

22

425

3

3

x

x

x

TMAT 103

§5.8

Equations with Fractions

§5.8 Equations with Fractions

• To solve an equation with fractions:1. Multiply both sides by the LCD2. 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

34

934 2 xx

41252

xxx

2Rfor Solve21 RQ

RQV