Warm-Up

6353 x

19357 xx

242 xx x = 6

x = 4

x = 6

(x + 5)(x + 5) x +5

x x2 5x

+5 5x 25x2 + 10x + 25

(a + b)(a + b) = a2 + 2ab + b2

Homework Check

Page 144, # 18 page 144, # 24

Page 148, # 24 page 148, # 30

Warm-up

Simplify the fraction

114

98. 2

45

60.

3 16 0. x2

Solve.

4 12 0. x x2

Standards for Rational Functions MM1A2c. Add, subtract, multiply, and divide

polynomials.

MM1A2d. Add, subtract, multiply, and divide rational expressions.

MM1A3d. Solve simple

rational equations that result

in linear equations or quadratic

equations with leading

coefficient of 1.

Rational Expressions

Essential questions:

1. What is a rational expression?

2. What doe the graph look like, and how does it move?

3. How do we simplify rational expressions?

11.3 Simplifying rational expressions

A fraction whose numerator and denominator are nonzero polynomials

Real Life Applications of Rational Expressional

Calculate concentrations Optimize perimeter / area Maximize profit / time Minimize Cost / Piece Average cost (cost / # pieces) Monthly payment per month Calculate piece part rates

Graphing Rational Expressions

Discuss domain, range, restrictions (has to be based on original expressions),

Graphs of Rational Expressions Have students graph the parent curve

X f(x) = 1/x Y

x can not be zero

Any restrictions on the domain?

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

f x = 1x

X f(x) = 1/x Y

5

0.5

0.2

0

-0.2

-0.5

-5

X f(x) = 1/x Y

5 1/5 0.2

0.5 1/0.5 2

0.2 1/0.2 5

0 1/0 ??

-0.2 -1/0.2 -5

-0.5 -1/0.5 -2

-5 -1/5 -0.2

Graphs of Rational Expressions Have students graph: f(x) = 5/(2x)

X f(x) = 1/2x YX f(x) = 1/(2x) Y

5

0.5

0.2

0

-0.2

-0.5

-5 Any restrictions on the domain?

x can not be zero

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

g x = 1

2x

f x = 1x

X f(x) = 1/(2x) Y

5 1/10 0.1

0.5 1/1 1

0.2 1/0.4 2.5

0 1/0 ??

-0.2 -1/0.4 -2.5

-0.5 -1/1 -1

-5 -1/10 -0.1

Graphs of Rational Expressions

Look at Geometer’s SketchPad for additional examples and explanation

Graphs of Rational ExpressionsSummary: Can not have values of x that make the

denominator equal zero (the domain is restricted)

Multiplying the parent function by a number x > 1 stretches graph vertically.

Multiplying the parent function by a number 0 < x < 1 shrinks graph vertically.

Adding a constant shifts the graph up or down

Multiplying by -1 reflects across the x-axis

Graphs of Rational ExpressionsHomework:

Pg 153, # 1 – 11 all

Simplifying Rational Expressions

Factor each numerator and denominator

Cross out the terms common to the numerator and denominator

Keep track of domain restrictions based on the original equation

When a rational expression’s numerator and denominator have no factors in common (other than 1)

1. Simplify a Rational Expression

18

60

2

3

x

x

3

10x

Reduce the numbers and subtract the exponents.

Where the larger

one is, is where the

leftovers go to keep

exponents positive

2. Simplify a Rational Expressionx x

x

2

2

6

3

Factor the top

x x

x

( )

6

3 2Cross out the common factor x.

( )x

x

6

3

3. Simplify a Rational Expression3

4 2

x

x x Factor the bottom

3

4

x

x x( ) Cross out the

common factor x.

3

4( ) x

4. Simplify a Rational Expression

5 10

15

2x x

x

Factor the top

5 1 2

15

x x

x

( )Cross out the

common factors of 5 and x.

( )1 2

3

x

5. Simplify a Rational Expression

x

x

2 16

3 12

Factor the top and bottom

( )( )

( )

x x

x

4 4

3 4Cross out the common factor (x + 4)

( )x 4

3

Divide Rational Expressions

(2x3 + 8x2 – 6x) ÷ 2x

x

xxx

2

682 23

x

xxx

2

)34(2 2

342 xx

Recognize Opposite Factors

When you have opposite factors, you will have to factor out a negative so that you can cancel.

6. Opposite Factors1

2 32

x

x x

Factor the bottom

( )

( )( )

1

3 1

x

x x

(1 – x) on the top and (x – 1) on the bottom are opposites. Factor out a negative so they will cancel.

( )

( )( )

x

x x

1

3 1 1

3( )x

7. Opposite Factorsx x

x

2 6 8

4

Factor the top

( )( )

( )

x x

x

4 2

4

(x – 4) on the top and (4 – x) on the bottom are opposites. Factor out a negative so they will cancel.

( )( )

( )

x x

x

4 2

4

( )( )

xx

2

11 2

x 2

Simplifying Rational ExpressionsSummary Review

Factor each numerator and denominator

Cross out the terms common to the numerator and denominator

Keep track of domain restrictions based on the original equation

Let’s practice!!

Pg 163, # 3 – 15 by 3’s and 16 – 19 all (9 problems)

Let: a = b

Multiply each side by b: ab = b2

Subtract a2 from each side

ab – a2 = b2 – a2

Factor each side: a(b – a) = (b – a)(b + a)

There is a (b – a) on each side, so cancel:

a = b + a

But a = b was given, so by substitution:

b + a = a + a = 2a

Giving a = 2a

Divide each side by a: 1 = 2

Warm-Up – What is wrong?

Have someone do problem 18 on page 163 (from last night’s homework) on the board.

What is the value of x to make the perimeter equal the area?

Multiply Rational Expressions

Factor both numerators and denominator

“Multiply” numerators “Multiply” denominators Cancel like factors in numerator and

denominator Discuss domain restrictions (has to be

based on original expressions),

Multiply Rational ExpressionsExample - Do what?

Factor

“Multiply”

Cancel

Answer – x can not

be -2, -3, or -4

127

3

2

232

2

xx

x

x

xx

)3)(4(

3

2

)1)(2(

xx

x

x

xx

)3)(4)(2(

)3)(1)(2(

xxx

xxx

)4(

)1(

x

x

Divide Rational Expressions Invert the divisor (turn up side down) Factor both numerators and

denominators “Multiply” numerators “Multiply” denominators Cancel like factors in numerator and

denominator Discuss domain, range, restrictions

(has to be based on original expressions),

)1)(2(

)2)(2)(1)(1(

xx

xxxx

Divide Rational ExpressionsExample - Do what?

Invert Divisor

Factor

Multiply

Cancel

Answer – x can not

be -1 or 2 or -2

4

1

2

12

2

x

x

x

x

)1(

)2)(2(

2

)1)(1(

x

xx

x

xx

)2)(1( xx

1

4

2

1 22

x

x

x

x

In Class Practice: Page 167, # 13 Page 167, # 16

Homework: Page 164, # 16 Page 167, # 3 – 18 by 3’s and 19 This is 8 problems

EOCT Review• Place to start: http://www.doe.k12.ga.us/• Click: Testing, End of Course Tests (also has

GHSGT reviews)• Math 1 Released Items gives sample

questions• Math 1 Released items commentary gives

answers and description to sample questions• Student study guides gives example

questions• EOCT Released Tests are for Algebra,

Geometry, not Math 1, but you can see what kind of questions they have asked in the past

Add / Subtract Rational Expressions

We must have the same denominator to be able to add or subtract rational expressions.

Can always find a common denominator by multiplying the denominators

Make common denominator by multiplying each rational expression by “1”

Add / Subtract Rational Expressions

The Least Common Denominator (LCD) of two or more rational expressions is the product of the factors in common (used only once) times the non-common factors.

Assuming these are denominators:Find the least common denominator

(2x – 4) and (x – 2) 2(x – 2) and (x – 2) Answer: 2(x – 2)

x2 – 5x and x2 – 3x – 10 x(x – 5) and (x – 5)(x + 2) Answer: x(x – 5)(x + 2)

Add / Subtract Rational Expressions

xx

x 2

8

14

Solve and state the restrictions

x can not be 0

8

82

8

14

xx

x

x

x

8

1614

xx

x

8

16

8

14

x

x

8

174

Add / Subtract Rational Expressions

1

2

12

xx

x

Solve and state the restrictions

Restrictions:

1

2

)1)(1(

xxx

x

1

1

)1(

2

)1)(1(

x

x

xxx

x

)1)(1(

)1(2

xx

xx

)1)(1(

22

xx

xx

)1)(1(

23

xx

x

x can not be1 or -1

Practice

Page 173, # 3 – 21 by 3’s and 22 and 23

Warm-Up

Divide by long division: 5889 ÷ 23

Divide a Polynomial by a Binomial

•It is easy if you can factor the numerator and denominator

Divide Polynomials (if you can not factor

Why do it?– To find factors, zeros, solutions, roots of the

equation The “zeros” are the values of x that

make the equation equal zero. Review long division Apply to polynomials If the value of x makes the expression

equal zero, you found a root. If the remainder = zero, you found a root.

Divide a polynomial by a binomial

Divide (4y by (y - 3)2 10 6y )

4 10 6

3

2y y

y

( )

First one goes on top, second one goes on bottom.

2 2 1 3

3

( )( )

( )

y y

y

GCF and Long factor the top

2 2 1( )y Cancel common factor (y – 3)

Divide Polynomials Fold paper hot dog Show calculations on one side Show division on the other side

Divide Polynomials Verify the one root, then find the

others (x3 – 2x2 – 5x + 6), x = -2

Roots are x = -2, 1, and 3 (x3 – 3x2 – 6x + 8), x = 4

Roots are x = -2, 1, and 4 (x3 – 11x2 + 7x + 147), x = -3

– Roots are x = -3 and 7 duplicity two (2x3 – 10x2 - 7x + 35), x = 5

Practice Pg 159, # 3 – 10 all (8 problems)

Solving Rational Equations Eliminate the denominator by

multiplying by the common denominator.

Solve: If x2, x, and constant:

– Move everything to one side of the equality sign, making everything equal zero

– Factor– Use Zero Product Rule to solve.– Check your answers

Painting a Car:

At a car body shop, Kayla needs 5 hours working alone to paint a car. It takes Emily 7 hours to paint the same car.

Write rational expressions that represent the fractional parts of the work that Kayla and Emily complete when painting the car together.

Painting a Car:

Write rational expressions that represent the fractional parts of the work that Kayla and Emily complete when painting the car together.

Kayla - amount painted per hour: x/5 Emily – amount painted per hour: x/7

Painting a Car:

Write and solve a rational equation to find the time it would take them to paint the car if they work together.

The time to paint one car is:

157

xx

Coordinate Geometry

Find any points of intersection for the graphs of the function and the function x

y8

2xy

Homework

Page 178, # 4 – 9 all and 13 – 15 all