6.1 Rational Expressions

Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial.

Examples of rational expressions

4 x 2x 2x – 5 x – 5

Domain = {x | x 0} Domain = {x | x 5/2} Domain = {x | x 5}

Graph of a Rational Function

y = 1 x

x y-2 -1/2-1 -1-1/2 -20 Undefined½ 21 12 ½

The graph does not cross the x = 0 line since x the graph is undefined there..The line x = 0 is called a vertical asymptote.

An Application: Modeling a train track curve.

Multiplication and Division of Rational Expressions

A • C = A 9x = 3B • C B 3x2 x

5y – 10 = 5 (y – 2) = 5 = 110y - 20 10 (y – 2) 10 2

2z2 – 3z – 9 = (2z + 3) (z – 3) = 2z + 3z2 + 2z – 15 (z + 5) (z – 3) z + 5

A2 – B2 = (A + B)(A – B) = (A – B)A + B (A + B)

Negation/Multiplying by –1

-y – 24y + 8

- = y + 2 4y + 8 OR -y - 2

-4y - 8

Examples

x3 – x x + 1x – 1 x

•

(x3 – x) (x + 1) x(x – 1)=

x (x2 – 1)(x + 1) x(x – 1)

=

= x (x + 1) (x – 1)(x + 1) x(x – 1)

= (x + 1)(x + 1) = (x + 1)2

x2 – 25 x2 –10x + 25x2 + 5x + 4 2x2 + 8x

=x2 – 25 2x2 + 8xx2 + 5x + 4 x2 –10x + 25

•

=(x + 5) (x – 5) • 2x(x + 4)(x + 4)(x + 1) • (x – 5) (x – 5)

=2x (x + 5)(x + 1)(x – 5)

Check Your Understanding

Simplify:

x2 –6x –7 x2 -1

Simplify:

1 3x - 2 x2 + x - 6

(x + 1) (x –7)(x + 1) (x – 1)

(x – 7)(x – 1)

1 x2 + x - 6x – 2 3•

1 (x + 3) (x – 2)x – 2 3

•

(x + 3) 3

6.2 Addition of Rational ExpressionsAdding rational expressions is like adding fractions

With LIKE denominators:

1 + 2 = 3 8 8 8

x + 3x - 1 = 4x - 1 x + 2 x + 2 x + 2

x + 2 (2 + x) (2 + x)3x2 + 4x - 4 3x2 + 4x -4 (3x2 + 4x – 4) (3x -2)(x + 2)

= =

= 1 (3x – 2)

Adding with UN-Like Denominators

3 + 14 8

(3) (2) + 18 8

6 + 18 8

7 8

1 + 2x2 – 9 x + 3

1 + 2(x + 3)(x – 3) (x + 3)

1 + 2 (x – 3)(x + 3)(x – 3) (x + 3)(x – 3)

1 + 2(x – 3) 1 + 2x – 6 2x - 5(x + 3) (x – 3) (x + 3) (x – 3) (x + 3) (x – 3)

= =

Subtraction of Rational Expressions

2x - x + 1x2 – 1 x2 - 1

To subtract rational expressions:Step 1: Get a Common DenominatorStep 2: Combine Fractions DISTRIBUTING the ‘negative sign’

BE CAREFUL!!

=2x – (x + 1)x2 -1

= x – 1(x + 1)(x –1)

= 1(x + 1)

= 2x – x - 1x2 -1

Check Your Understanding

Simplify:

b b-12b - 4 b-2-

b b-12(b – 2) b-2

-

b -b+12(b – 2) b-2

+

b2(b – 2)

2(-b+1)2(b – 2)

+

b –2b+22(b – 2)

-b + 22(b – 2)= =

-1(b – 2)2(b – 2)

= -12

6.3 Complex Fractions

A complex fraction is a rational expression that contains fractions in its numerator, denominator, or both.

Examples:

15

47

xx2 – 16

1x - 4

1x

+ 2x2

3x

- 1x2

7/20 xx + 4

x + 23x - 1

6.4 Division by a Monomial3x2 + x 5x3 – 15x2

x 15x

4x2 + 8x – 12 5x2y + 10xy2

4x2 5xy

15A2 – 8A2 + 12 12A5 – 8A2 + 12 4A 4A

Polynomial Long DivisionExample: Divide 4 – 5x – x2 + 6x3 by 3x – 2.

Begin by writing the divisor and dividend in descending powers of x. Then, figure out how many times 3x divides into 6x3.

3x – 2 6x3 – x2 – 5x + 42x2 Divide: 6x3/3x = 2x2.

6x3 – 4x2

Multiply.

Multiply: 2x2(3x – 2) = 6x3 – 4x2.

Divide: 3x2/3x = x.

Now, divide 3x2 by 3x to obtain x, multiply then subtract.

3x – 2 6x3 – x2 – 5x + 46x3 – 4x2

2x2 + x

3x2 – 5x

Multiply.

Multiply: x(3x – 2) = 3x2 – 2x.

3x2 – 2xSubtract 3x2 – 2x from 3x2 – 5x and bring down 4.

+ 4-3x

– 5x Subtract 6x3 – 4x2 from 6x3 – x2 and bring down –5x.

3x2

Subtract -3x + 2 from -3x + 4, leaving a remainder of 2.

2

-3x +2

-1

Answer: 2x2 + x – 1 + 2 3x - 2

3x -11

3x3 + 9x2 + 9x

-11x2 - 5x - 3

-11x2 - 33x - 33

28x+30

More Long Division

6.5-6.6 Rational Equations

3x = 3 x + 1 = 3 6 = x2x – 1 x – 2 x - 2 x + 1

(2x – 1)

3x = 3(2x – 1)3x = 6x – 3-3x = -3

x = 1

(x - 2)

x + 1 = 3

x = 2

(x + 1)

6 = x (x + 1)

6 = x2 + x

x2 + x – 6 = 0

(x + 3 ) (x - 2 ) = 0

x = -3 or x = 2Careful! – What doYou notice about theanswer?

Rational Equations Cont…To solve a rational equation:

Step 1: Factor all polynomialsStep 2: Find the common denominatorStep 3: Multiply all terms by the common denominatorStep 4: Solve

x + 1 - x – 1 = 1 2x 4x 3

(12x)

= 6 (x + 1) -3(x – 1) = 4x6x + 6 –3x + 3 = 4x

3x + 9 = 4x -3x -3x 9 = x

Other Rational Equation Examples

3 + 5 = 12x – 2 x + 2 x2 - 4

3 + 5 = 12x – 2 x + 2 (x + 2) (x – 2)

(x + 2)(x – 2)

3(x + 2) + 5(x – 2) = 12

3x + 6 + 5x – 10 = 12

8x – 4 = 12 + 4 + 4

8x = 16

x = 2

1 + 1 = 3x x2 4

(4x2)

4x + 4 = 3x2

3x2 - 4x - 4 = 0

(3x + 2) (x – 2) = 0

3x + 2 = 0 or x – 2 = 0

3x = -2 or x = 2

x = -2/3 or x = 2

Check Your Understanding

Simplify:x 1x2 – 1 x2 – 1

1 3x – 2 x

1 1 2x(x – 1) x2 – 1 x(x + 1)

Solve6 1x 2

3 22x – 1 x + 1

2 3 xx – 1 x + 2 x2 + x - 2

+

-

+ -

- = 1

=

+ =

1x - 1

2(x – 3)x(x – 2)

3x(x – 1)(x + 1)

4

5

-1/4

1 = 1 + 1F p q

Solve for p:Try this one:

6.7 Proportions & VariationProportion equality of 2 ratios. Proportions are used to solve problems in everyday life.

1. If someone earns $100 per day, then how many dollars can the person earn in 5 days?

100 x (x)(1) = (100)(5) 1 5 x = 500

2. If a car goes 210 miles on 10 gallons of gas, the car can go 420 miles on X gallons

210 420 (210)(x) = (420)(10) 10 x (210)(x) = 4200

x = 4200 / 210 = 20 gallons

3. If a person walks a mile in 16 min., that person can walk a half mile in x min.

16 x (x)(1) = ½(16) 1 ½ x = 8 minutes

=

=

=

The Shadow ProblemJuan is 6 feet tall, but his shadow is only 2 ½ feet long.There is a tree across the street with a shadow of 100 feet.The sun hits the tree and Juan at the same angle to make the shadows.How tall is the tree?

6ft

2 ½ ft

x

100 ft

6x

= 2.5100

2.5x = (100)(6)

2.5x = 6002.5 2.5

x = 240 feet

personheight treeheight

personshadow

treeshadow

7.6 Direct Variationy = kx y is directly proportional to x.

y varies directly with xk is the constant of proportionality

Example: y = 9x (9 is the constant of proportionality)Let y = Your payLet x – Number of Hours workedYour pay is directly proportional to the number of hours worked.

Example1:

Salary (L) varies directly as the number of hours worked (H). Write an equation that expresses this relationship.

Salary = k(Hours)

L = kH

Example 2:

Aaron earns $200 after working 15 hours.

Find the constant of proportionality using

your equation in example1..

200 = k(15)

So, k = 200/15 = 13.33

Inverse Variation

y = k y is inversely proportional to x x y varies inversely as x

Example: y varies inversely with x.If y = 5 when x = 4, find the constant of proportionality (k)

5 = k So, k = 20 4

Direct Variation with Power

y = kxn

y is directly proportional to the nth power of x

Example: Distance varies directly as the square of the time (t)

Distance = kt2

D = kt2

Joint Variation

y = kxp • y varies jointly as x and p