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Chapter 6. Rational Expressions. Rational Functions and Multiplying and Dividing Rational Expressions. § 6.1. Rational Expressions. Rational expressions can be written in the form where P and Q are both polynomials and Q  0. Examples of Rational Expressions.

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Rational expressions

Chapter 6

Rational Expressions


Rational functions and multiplying and dividing rational expressions

Rational Functions and Multiplying and Dividing

Rational Expressions

§ 6.1


Rational expressions1

Rational Expressions

Rational expressions can be written in the form where P and Q are both polynomials and Q 0.

Examples of Rational Expressions


Rational expressions

Evaluating Rational Expressions

To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result.

Example:

Evaluate the following expression for y = 2.


Rational expressions

Evaluating Rational Expressions

In the previous example, what would happen if we tried to evaluate the rational expression for y = 5?

This expression is undefined!


Rational expressions

Undefined Rational Expressions

We have to be able to determine when a rational expression is undefined.

A rational expression is undefined when the denominator is equal to zero.

The numerator being equal to zero is okay (the rational expression simply equals zero).


Rational expressions

Undefined Rational Expressions

Find any real numbers that make the following rational expression undefined.

Example:

The expression is undefined when 15x + 45 = 0.

So the expression is undefined when x = 3.


Rational expressions

For any rational expression and any polynomial R, where R ≠ 0,

or, simply,

Simplifying Rational Expressions

Simplifying a rational expression means writing it in lowest terms or simplest form.

Fundamental Principle of Rational Expressions


Rational expressions

Simplifying Rational Expressions

Simplifying a Rational Expression

1) Completely factor the numerator and denominator of the rational expression.

2) Divide out factors common to the numerator and denominator. (This is the same thing as “removing the factor of 1.”)

Warning!

Only common FACTORS can be eliminated from the numerator and denominator. Make sure any expression you eliminate is a factor.


Rational expressions

Simplifying Rational Expressions

Example:

Simplify the following expression.


Rational expressions

Simplifying Rational Expressions

Example:

Simplify the following expression.


Rational expressions

Simplifying Rational Expressions

Example:

Simplify the following expression.


Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions

The rule for multiplying rational expressions is

as long as Q  0 and S  0.

1) Completely factor each numerator and denominator.

2) Use the rule above and multiply the numerators and denominators.

3) Simplify the product by dividing the numerator and denominator by their common factors.


Multiplying rational expressions1

Multiplying Rational Expressions

Example:

Multiply the following rational expressions.


Multiplying rational expressions2

Multiplying Rational Expressions

Example:

Multiply the following rational expressions.


Dividing rational expressions

Dividing Rational Expressions

Dividing Rational Expressions

The rule for dividing rational expressions is

as long as Q  0 and S  0 and R  0.

To divide by a rational expression, use the rule above and multiply by its reciprocal. Then simplify if possible.


Dividing rational expressions1

Dividing Rational Expressions

Example:

Divide the following rational expression.


Rational expressions

§ 6.2

Adding and Subtracting Rational Expressions


Adding and subtracting with common denominators

Adding and Subtracting with Common Denominators

The rule for multiplying rational expressions is

Adding or Subtracting Rational Expressions with Common Denominators

are rational expression, then


Adding rational expressions

Add the following rational expressions.

Adding Rational Expressions

Example:


Subtracting rational expressions

Subtract the following rational expressions.

Subtracting Rational Expressions

Example:


Subtracting rational expressions1

Subtract the following rational expressions.

Subtracting Rational Expressions

Example:


Least common denominators

To add or subtract rational expressions with unlike denominators, you have to change them to equivalent forms that have the same denominator (a common denominator).

This involves finding the least common denominator of the two original rational expressions.

Least Common Denominators


Least common denominators1

Finding the Least Common Denominator

1) Factor each denominator completely,

2) The LCD is the product of all unique factors each raised to a power equal to the greatest number of times that the factor appears in any factored denominator.

Least Common Denominators


Least common denominators2

Find the LCD of the following rational expressions.

Least Common Denominators

Example:


Least common denominators3

Find the LCD of the following rational expressions.

Least Common Denominators

Example:


Least common denominators4

Find the LCD of the following rational expressions.

Least Common Denominators

Example:


Least common denominators5

Find the LCD of the following rational expressions.

Least Common Denominators

Example:

Both of the denominators are already factored.

Since each is the opposite of the other, you can

use either x – 3 or 3 – x as the LCD.


Multiplying by 1

To change rational expressions into equivalent forms, we use the principal that multiplying by 1 (or any form of 1), will give you an equivalent expression.

Multiplying by 1


Equivalent expressions

Rewrite the rational expression as an equivalent rational expression with the given denominator.

Equivalent Expressions

Example:


Unlike denominators

As stated in the previous section, to add or subtract rational expressions with different denominators, we have to change them to equivalent forms first.

Unlike Denominators


Unlike denominators1

Adding or Subtracting Rational Expressions with Unlike Denominators

Find the LCD of the rational expressions.

Write each rational expression as an equivalent rational expression whose denominator is the LCD found in Step 1.

Add or subtract numerators, and write the result over the denominator.

Simplify resulting rational expression, if possible.

Unlike Denominators


Adding with unlike denominators

Add the following rational expressions.

Adding with Unlike Denominators

Example:


Subtracting with unlike denominators

Subtract the following rational expressions.

Subtracting with Unlike Denominators

Example:


Subtracting with unlike denominators1

Subtract the following rational expressions.

Subtracting with Unlike Denominators

Example:


Adding with unlike denominators1

Add the following rational expressions.

Adding with Unlike Denominators

Example:


Rational expressions

§ 6.3

Simplifying Complex Fractions


Complex rational fractions

Complex Rational Fractions

Complex rational expressions (complex fraction) are rational expressions whose numerator, denominator, or both contain one or more rational expressions.

There are two methods that can be used when simplifying complex fractions.


Simplifying complex fractions

Simplifying a Complex Fraction(Method 1)

Simplify the numerator and the denominator of the complex fraction so that each is a single fraction.

Perform the indicated division by multiplying the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction.

Simplify, if possible.

Simplifying Complex Fractions


Simplifying complex fractions1

Simplifying Complex Fractions

Example:


Simplifying complex fractions2

Simplifying a Complex Fraction(Method 2)

Multiply the numerator and the denominator of the complex fraction by the LCD of the fractions in both the numerator and denominator.

Simplify, if possible.

Simplifying Complex Fractions


Simplifying complex fractions3

Simplifying Complex Fractions

Example:


Simplifying with negative exponents

When you have a rational expression where some of the variables have negative exponents, rewrite the expression using positive exponents.

The resulting expression is a complex fraction, so use either method to simplify the expression.

Simplifying with Negative Exponents


Simplifying with negative exponents1

Simplifying with Negative Exponents

Example:


Rational expressions

§ 6.4

Dividing Polynomials: Long Division and Synthetic Division


Dividing a polynomial by a monomial

Dividing a Polynomial by a Monomial

Dividing a Polynomial by a Monomial

Divide each term in the polynomial by the monomial.

Example: Divide


Dividing a polynomial by a monomial1

Dividing a Polynomial by a Monomial

Example: Divide


Dividing polynomials

Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide.

Dividing Polynomials


Dividing polynomials1

Dividing Polynomials

We then write our result as

Divide 43 into 72.

Multiply 1 times 43.

Subtract 43 from 72.

Bring down 5.

Divide 43 into 295.

Multiply 6 times 43.

Subtract 258 from 295.

Bring down 6.

Divide 43 into 376.

Multiply 8 times 43.

Subtract 344 from 376.

Nothing to bring down.


Rational expressions

Dividing Polynomials

As you can see from the previous example, there is a pattern in the long division technique.

Divide

Multiply

Subtract

Bring down

Then repeat these steps until you can’t bring down or divide any longer.

We will incorporate this same repeated technique with dividing polynomials.


Dividing polynomials2

Dividing Polynomials

-

35

x

-

15

Divide 7x into 28x2.

Multiply 4x times 7x+3.

Subtract 28x2 + 12x from 28x2 – 23x.

Bring down – 15.

Divide 7x into –35x.

Multiply – 5 times 7x+3.

Subtract –35x–15 from –35x–15.

Nothing to bring down.

So our answer is 4x – 5.


Rational expressions

-

-

10

10

2

2

x

x

+

-

+

2

2

x

7

4

x

6

x

8

-

20

x

-

-

20

x

70

+

8

2

+

14

x

4

x

78

+

We write our final answer as

+

(

2

x

7

)

Dividing Polynomials

Divide 2x into 4x2.

Multiply 2x times 2x+7.

Subtract 4x2 + 14x from 4x2 – 6x.

Bring down 8.

Divide 2x into –20x.

Multiply -10 times 2x+7.

Subtract –20x–70 from –20x+8.

78

Nothing to bring down.


Dividing polynomials using long division

Dividing Polynomials Using Long Division

Example: Divide:

1. Divide the leading term of the dividend, c2, by the first term of the divisor, x.

2. Multiply c by c + 1.

3. Subtract c2 + c from c2 + 3c – 2.

Continued.


Dividing polynomials using long division1

Dividing Polynomials Using Long Division

2

c

Remainder

Example continued:

Bring down the next term to obtain a new polynomial.

4. Repeat the process until the degree of the remainder is less than the degree of the binomial divisor.

5. Check by verifying that (Quotient)(Divisor) + Remainder = Dividend.


Dividing polynomials using long division2

Dividing Polynomials Using Long Division

No remainder

Example: Divide: (y2 – 5y + 6) ÷ (y – 2)

y

– 3

y2 – 2y

+ 6

– 3y

– 3y + 6

0

Check : (y – 2)(y – 3)

= y2 – 5y + 6

(y2 – 5y + 6)÷ (y – 2) = y – 3


Synthetic division

Synthetic Division

To find the quotient and remainder when a polynomial of degree 1 or higher is divided by x – c, a shortened version of long division called synthetic division may be used.

Long Division

Synthetic Division

Subtraction signs are not necessary.

The original terms in red are not needed because they are the same as the term directly above.

Continued.


Synthetic division1

Synthetic Division

The variables are not necessary.

Continued.

The “boxed” numbers can be aligned horizontally.

Continued.


Synthetic division2

Synthetic Division

The first two numbers in the last row are the coefficients of the quotient, the last number is the remainder.

To simplify further, the top row can be removed and instead of subtracting, we can change the sign of each entry and add.

The leading coefficient of the dividend can be brought down.

The x + 1 is replaced with a – 1.

Quotient

Remainder

Continued.


Rational expressions

§ 6.5

Solving Equations Containing Rational Expressions


Solving equations

Solving Equations

First note that an equation contains an equal sign and an expression does not.

To solve EQUATIONS containing rational expressions, clear the fractions by multiplying both sides of the equation by the LCD of all the fractions.

Then solve as in previous sections.

Note: this works for equations only, not simplifying expressions.


Solving equations1

Solving Equations

Example:

Solve the following rational equation.

Check in the original equation.

true


Solving equations2

Solving Equations

Example:

Solve the following rational equation.

Continued.


Rational expressions

So the solution is

Solving Equations

Example continued:

Substitute the value for x into the original equation, to check the solution.

true


Solving equations3

Solving Equations

Example:

Solve the following rational equation.

Continued.


Rational expressions

So the solution is

Solving Equations

Example continued:

Substitute the value for x into the original equation, to check the solution.

true


Solving equations4

Solving Equations

Example:

Solve the following rational equation.

Continued.


Rational expressions

Solving Equations

Example continued:

Substitute the value for x into the original equation, to check the solution.

true

So the solution is x = 3.


Solving equations5

Solving Equations

Example:

Solve the following rational equation.

Continued.


Rational expressions

Solving Equations

Example continued:

Substitute the value for x into the original equation, to check the solution.

Since substituting the suggested value of a into the equation produced undefined expressions, the solution is .


Rational expressions

§ 6.6

Rational Equations and Problem Solving


Rational expressions

Solving Equations for a Specified Variable

Clear the equation of fractions or rational expressions by multiplying each side of the equation by the LCD of the denominators in the equation.

Use the distributive property to remove grouping symbols such as parentheses.

Combine like terms on each side of the equation.

Use the addition property of equality to rewrite the equation as an equivalent equation with terms containing the specified variable on one side and all other terms on the other side.

Use the distributive property and the multiplication property of equality to get the specified variable alone.

Solving Equations with Multiple Variables


Rational expressions

Solving Equations with Multiple Variables

Example:

Solve the following equation for R1


Finding an unknown number

Finding an Unknown Number

Read and reread the problem. If we let

n = the number, then

= the reciprocal of the number

Example:

The quotient of a number and 9 times its reciprocal is 1. Find the number.

1.) Understand

Continued


Rational expressions

The quotient of

is

1

a number

and 9 times its reciprocal

n

=

1

Finding an Unknown Number

Example continued:

2.) Translate

Continued


Rational expressions

Finding an Unknown Number

Example continued:

3.) Solve

Continued


Rational expressions

Finding an Unknown Number

Example continued:

4.) Interpret

Check: We substitute the values we found from the equation back into the problem. Note that nothing in the problem indicates that we are restricted to positive values.

true

true

State: The missing number is 3 or –3.


Ratios and rates

Ratios and Rates

Ratio is the quotient of two numbers or two quantities.

The ratio of the numbers a and b can also be written as a:b, or .

The units associated with the ratio are important.

The units should match.

If the units do not match, it is called a rate, rather than a ratio.


Proportions

Proportion is two ratios (or rates) that are equal to each other.

Proportions

We can rewrite the proportion by multiplying by the LCD, bd.

This simplifies the proportion to ad = bc.

This is commonly referred to as the cross product.


Solving proportions

Solve the proportion for x.

Solving Proportions

Example:

Continued.


Rational expressions

So the solution is

Solving Proportions

Example continued:

Substitute the value for x into the original equation, to check the solution.

true


Solving proportions1

If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh?

Solving Proportions

Example:


Solving proportions2

Given the following prices charged for various sizes of picante sauce, find the best buy.

10 ounces for $0.99

16 ounces for $1.69

30 ounces for $3.29

Solving Proportions

Example:

Continued.


Solving proportions3

Solving Proportions

Example continued:

Size Price Unit Price

10 ounces $0.99 $0.99/10 = $0.099

16 ounces $1.69 $1.69/16 = $0.105625

30 ounces $3.29 $3.29/30  $0.10967

The 10 ounce size has the lower unit price, so it is the best buy.


Solving a work problem

Solving a Work Problem

Read and reread the problem. By using the times for each roofer to complete the job alone, we can figure out their corresponding work rates in portion of the job done per hour.

Time in hrs

Portion job/hr

Experienced roofer 26 1/26

Beginner roofer 39 /39

Together t 1/t

Example:

An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together?

1.) Understand

Continued


Rational expressions

Solving a Work Problem

Example continued:

2.) Translate

Since the rate of the two roofers working together would be equal to the sum of the rates of the two roofers working independently,

Continued


Rational expressions

Solving a Work Problem

Example continued:

3.) Solve

Continued


Rational expressions

Solving a Work Problem

Example continued:

4.) Interpret

Check: We substitute the value we found from the proportion calculation back into the problem.

true

State: The roofers would take 15.6 hours working together to finish the job.


Solving a rate problem

Solving a Rate Problem

Read and reread the problem. By using the formula d=rt, we can rewrite the formula to find that t = d/r.

We note that the rate of the boat downstream would be the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current.

Distance rate time = d/r

Down 20 r + 5 20/(r + 5)

Up 10 r – 5 10/(r – 5)

Example:

The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water.

1.) Understand

Continued


Rational expressions

Solving a Rate Problem

Example continued:

2.) Translate

Since the problem states that the time to travel downstairs was the same as the time to travel upstairs, we get the equation

Continued


Rational expressions

Solving a Rate Problem

Example continued:

3.) Solve

Continued


Rational expressions

Solving a Rate Problem

Example continued:

4.) Interpret

Check: We substitute the value we found from the proportion calculation back into the problem.

true

State: The speed of the boat in still water is 15 mph.


Rational expressions

§ 6.7

Variation and Problem Solving


Direct variation

Direct Variation

y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that y = kx.

The family of equations of the form y = kx are referred to as direct variation equations.

The number k is called the constant of variation or the constant of proportionality.


Direct variation1

If y varies directly as x, find the constant of variation k and the direct variation equation, given that y = 5 when x = 30.

y = kx

5 = k·30

k = 1/6

Direct Variation

So the direct variation equation is

1

6

y =

x


Direct variation2

If y varies directly as x, and y = 48 when x = 6, then find y when x = 15.

y = kx

48 = k·6

8 = k

So the equation is y = 8x.

y = 8·15

y = 120

Direct Variation

Example:


Direct variation3

At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer.If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places.

Direct Variation

Example:

Continued.


Direct variation4

Direct Variation

Example continued:

We substitute our given value for the elevation into the equation.

So our equation is


Inverse variation

y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that y = k/x.

The family of equations of the form y = k/x are referred to as inverse variation equations.

The number k is still called the constant ofvariation or the constant of proportionality.

Inverse Variation


Inverse variation1

If y varies inversely as x, find the constant of variation k and the inverse variation equation, given that y = 63 when x = 3.

y = k/x

63 = k/3

k = 63·3

k = 189

Inverse Variation

189

x

So the inverse variation equation is

y =

Example:


Powers of x

y can vary directly or inversely as powers of x, as well.

y varies directly as a power of x if there is a nonzero constant k and a natural number n such that y = kxn

Powers of x

y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that


Powers of x1

The maximum weight that a circular column can hold is inversely proportional to the square of its height.

If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold.

Powers of x

Example:

Continued.


Powers of x2

Powers of x

Example continued:

We substitute our given value for the height of the column into the equation.

So our equation is


Combined variation

y varies jointly as, or is jointly proportional to two (or more) variables x and z, if there is a nonzero constant k such that y = kxz.

k is STILL the constant of variation or the constant of proportionality.

We can also use combinations of direct, inverse, and joint variation. These variations are referred to as combined variations.

Combined Variation


Combined variation1

Write the following statements as equations.

a varies jointly as b and c

a = kbc

P varies jointly as R and the square of S

P = kRS2

The weight (w) varies jointly with the width (d) and the square of the height (h) and inversely with the length (l)

Combined Variation

Example:


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