Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression...

Section 6.1

Rational Expressions

OBJECTIVES

A Find the numbers that make a rational expression undefined.

OBJECTIVES

B Write an equivalent fraction with the indicated denominator.

OBJECTIVES

C Write a fraction in the standard forms.

OBJECTIVES

D Reduce a fraction to lowest terms.

DEFINITION

If P and Q are polynomials:

Rational Expressions

( 0)P

QQ

DEFINITION

The variables in a rational expression may not be replaced by values that will make the denominator zero.

Undefined Rational Expressions

DEFINITION

If P, Q, and K are polynomials

PQ

= P • KQ • K

Fundamental Property of Fractions

Reducing FractionsPROCEDURE

1. Write numerator and denominator in factored form.

2. Find the GCF.

Reducing FractionsPROCEDURE

3. Replace the quotient of the common factors by 1.

4. Rewrite in lowest terms.

DEFINITION

Quotient of Additive Inverses

a – bb – a

= –1

Practice Test

Exercise #1

Chapter 6Section 6.1A,B

Find the undefined value(s) for

a. x – 2

3x + 4 is undefined when:

3x = – 4

x = – 4

3

3x + 4 = 0

Write the fraction with the indicated denominator.

b. 2x 2

9y 4= ?

36y7

36y7= 9y 4 • 4y3

2x 2 • 4y3

9y 4 • 4y3 =

8x 2y3

36y 7

Practice Test

Exercise #2

Chapter 6Section 6.1C

Write in standard form

a. – –5

y

=

5

y

Write in standard form

b. –

x – y

5

= –x + y

5

= y – x

5

Practice Test

Exercise #4

Chapter 6Section 6.1D

Reduce to lowest terms.

y2 – x2

x3 – y3

2 2

3 3

–1 –=

–

x y

x y

2 2

– – +=

– + +

x y x y

xy yx y x

Factor out – 1

Difference of Squares

Difference of Cubes

Reduce to lowest terms.

2 2

– – +=

– + +

x y x y

xy yx y x

2 2

– +=

+ +

x y

xy yx

Section 6.2

Multiplication and Division of Rational Expressions

OBJECTIVES

A Multiply rational expressions.

OBJECTIVES

B Divide rational expressions.

OBJECTIVES

C Use multiplication and division together.

DEFINITION

Multiplication of Rational Expressions

ab

• cd = a • c

b • d

( 0 0) b , d

To Multiply Rational Expressions

PROCEDURE

1. Factor the numerators and denominators completely.

2. Simplify each expression.

To Multiply Rational Expressions

PROCEDURE

3. Multiply remaining factors.

4. The final product should be in lowest terms.

DEFINITION

Division of Real Numbers

÷a c a d =b d b c •

( and 0) b, d, c

Practice Test

Exercise #6

Chapter 6Section 6.2B

Perform the indicated operations.

2 – xx + 3

÷ x3 – 8x + 5

x3 + 27

x + 5

3

3– – 2 + 5 + 27

= + 3 + 5– 8

x x x

x xx

2– 2 + 2 + 4x x x

2+ 3 – 3x + 9x x

2

2– – 3 + 9

= + 2 + 4

x x

x x

2

2

+ 3 – 3x + 9– – 2 + 5=

+ 3 + 5– 2 + 2 + 4

x xx xx xx x x


Practice Test

Exercise #7



2 – xx + 3

÷ x3 – 8x + 5

x3 + 27

x + 5

3

3– – 2 + 5 + 27

= + 3 + 5– 8

x x x

x xx

2– 2 + 2 + 4x x x

2+ 3 – 3x + 9x x

2

2– – 3 + 9

= + 2 + 4

x x

x x

2

2

+ 3 – 3x + 9– – 2 + 5=

+ 3 + 5– 2 + 2 + 4

x xx xx xx x x


Section 6.3

Addition and Subtraction of Rational Expressions

OBJECTIVES

A Add or subtract rational expressions with the same denominator.

OBJECTIVES

B Add or subtract rational expressions with different denominators.

Finding the LCD of Two or More Rational Expressions

PROCEDURE

1.Factor denominators. Place factors in columns.(Not necessary to factor monomials).


PROCEDURE

2.Select the factor with the greatest exponent from each column.


PROCEDURE

3.The product of all the factors obtained is the LCD.

To Add or Subtract Fractions with Different Denominators.

PROCEDURE

1.Find the LCD.

2. Write all fractions as equivalent ones with LCDas denominator.

To Add or Subtract Fractions with Different Denominators.

PROCEDURE

3.Add numerators.

4. Simplify.

Practice Test

Exercise #9a



a.

x +1

x 2 + x – 2 +

x + 4

x 2 – 1


a.

x +1

x 2 + x – 2 +

x + 4

x 2 – 1

+1 + 4= +

+ 2 – 1 +1 – 1

x x

x x x x

= + 2 – 1 +1LCD x x x

+1 + 4 = +

+ 2 – 1 +1 – 1

x x

x x x x

= + 2 – 1 +1x x x

2 2+ 2 + 1 + + 6 + 8

= + 2 – 1 +1

x x x xx x x

+ 1x

+ 1x

+ 2x

+ 2x


= + 2 – 1 +1LCD x x x

2 + 2 +1x x 2+ + 6 + 8x x

2 2 + + 2 + 6 +1+ 8

= + 2 – 1 +1

x x x xx x x

22 + 8 + 9

= + 2 – 1 +1

x xx x x


Section 6.4

Complex Fractions

OBJECTIVES

A Write a complex fraction as a simple fraction in reduced form.

Simplifying Complex FractionsPROCEDURE

Multiply the numerator and denominator of the complex fraction by the LCD of all simple fractions.

METHOD 1

PROCEDURE

Perform operations indicated in numerator and denominator.Then divide numerator by denominator.

Simplifying Complex Fractions

METHOD 2

Practice Test

Exercise #10

Chapter 6Section 6.4A

Simplify.

x + 1

x2

x – 1

x3

Multiply by LCD

Simplify.

2

3

3

3

1 + •

• =

1 –

xx

xx

x

x

=

x 4 + x

x 4 – 1

3

2 2

+1 =

+1 – 1

x x

x x

2

2

+1 =

+1 1

x x x

x x

3

2 2

+1 =

+1 – 1

x x

x x

2

2

+1 – +1 =

+1 +1 – 1

x x x x

x x x

Simplify.

Section 6.5

Division of Polynomials and Synthetic Division

OBJECTIVES

A Divide a polynomial by a monomial.

OBJECTIVES

B Use long division to divide one polynomial by another.

OBJECTIVES

C Completely factor a polynomial when one of the factors is known.

OBJECTIVES

D Use synthetic division to divide one polynomial by a binomial.

OBJECTIVES

E Use the remainder theorem to verify that a number is a solution of a given equation.

Dividing a Polynomial by a Monomial

RULE

Divide each term in the polynomial by the monomial.

DEFINITION

The Remainder Theorem

If P(x) is divided by x –k , then the remainder is P(k).

DEFINITION

The Factor Theorem

When P(x) has a factor (x –k) , it means that P(k) = 0.

Practice Test

Exercise #13


Divide.

3 2 – – 4 ÷ 2 + 26x x x

Write in descending order.

3 22 – 4 – ÷ 2+ + 20 6x x xx

Use 0x2 for missing term.

– 2x – 2

– 4

2x+ 2 2x3 + 2x 2

– 2x 2

– 2x

– 2x 2 – 2x

–x –1Divide.

x 2

– 4x

– 6

Remainder

2x3 + 0x 2 – 4x – 6

Practice Test

Exercise #14


2x3 + 3x 2 – 23x – 12 x – 3

2x3 – 6x 2

9x2 – 23x

4x – 12

9x2 – 27x

Factor 2x3 + 3x2 – 23x – 12 if x – 3 is one of its factors.

2x 2

4x – 12

0

+ 9x + 4

Factor 2x3 + 3x2 – 23x – 12 if x – 3 is one of its factors.

Factor 2x2 + 9x + 4

= 2x + 1 x + 4

– 3 2x + 1 x + 4 x

Factors of 2x3 + 3x2 – 23x – 12 are :

Practice Test

Exercise #16

Chapter 6Section 6.5E

Use synthetic division to show that – 1 is a solution of

x4 – 4x3 – 7x2 + 22x + 24 = 0

–1 1 –4 –7 22 24

–1 is a solution of the equation since the remainder R=0.

–24 +2

–1 +5

(0) 1 –5 –2 24

Section 6.6

Equations Involving Rational Expressions

OBJECTIVES

A Solve equations involving rational expressions.

OBJECTIVES

B Solve applications using proportions.

Solving Equations Containing Rational Expressions

PROCEDURE

1. Factor denominators and multiply both sides of the equation by the LCD.

PROCEDURE

2. Write the result in reduced form. Use the distributive property to remove parentheses.


PROCEDURE

3. Determine whether the equation is linear or quadratic and solve accordingly.


PROCEDURE

4. Check that the proposed solution satisfies the equation. If not, discard it as an extraneous solution.


DEFINITION

Property of Proportions

If (where 0),

then

a c = b, db d

a d = b c

• •

A proportion is true if the cross products are equal.

Practice Test

Exercise #18


Solve: 18x – 2 – 3x – 1 = 1

18

x2–

3 x

= 1

x – 2 =

1

x2, x – 1 =

1x

• x2 • x2 • x2

18 – 3x = x2

18

x2–

3 x

= 1 • x2 • x2 • x2

1

x

x = – 6 or x = 3

Solve: 18x – 2 – 3x – 1 = 1

0 = x2 + 3x – 18OO

0 = (x + 6)(x – 3)FF

or x – 3 = 0 x + 6 = 0FF

Practice Test

Exercise #19


A recipe for curried shrimp that normally serves four was once served to 200 guests at a wedding reception. One of the ingredients in the recipe is11

2 cups of chicken broth.

a. How much chicken broth was required to make the recipe for 200 people?

People

Broth =

4

112

= 200x

4x = 200 •

32

4x = 300

x =

3004

a.

= 75 cups

b. If a medium-sized can of chicken broth contains 2 cups of broth, how many cans are necessary?

75 cups

2 cups

= 37

12

38 cans

Section 6.7

Applications: Problem Solving

OBJECTIVES

A Solve integer problems.

OBJECTIVES

B Solve work problems.

OBJECTIVES

C Solve distance problems.

OBJECTIVES

D Solve for a specified variable.

PROCEDURE:

Read Select Think Use Verify

RSTUV Method for Solving Word Problems

Practice Test

Exercise #21


Jack can mow the lawn in 4 hours and Jill can mow it in 3. How long would it take them to mow the lawn if they work together?

let x = Time working together (hr)

Time workedTime working alone

= amount done

Jack does

x4

of the work

Jill does

x3

of the work

Together they do 1 full job

x4

+ x3

= 1

3x + 4x = 12

7x = 12


x =

127

or 157

It takes 1

57

hours if they work together.


Section 6.8

Variation

OBJECTIVES

A Direct variation.

OBJECTIVES

B Inverse variation.

OBJECTIVES

C Joint variation.

OBJECTIVES

D Solve applications involving direct, inverse, and joint variation.

DEFINITION

Direct Variation

y varies directly as x if there is a constant k:

y = kx

DEFINITION

Inverse Variation

y varies inversely as x if there is a constant k:

y = kx

DEFINITION

Joint Variation

z varies jointly with x and y if there is a constant k:

z = kxy

Practice Test

Exercise #24


C is directly proportional to m.

a. Write an equation of variation

with k as the constant.

b. Find k when C = 12 and m =

1

3 .


a. Write an equation of variation

with k as the constant.

C = km

Direct Variation y = kx


C = km

12 = k

13

k = 36

b. Find k when C = 12 and m =

1

3 .

Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression...

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