What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin...

What is a rational expression?It is a ratio of two polynomial expressions, like this:

We will begin by reducing fractions

Now we will reduce Polynomials

Simplify rational expressions means that we could not reduce or factor anything else out of the expression.

Simplify, Multiply, and DivideRational Expressions

Now let’s reduce Polynomials

When dividing polynomials, they are called ____rational expressions____

There are two steps for reducing/simplifying rational expressions.

Step 1. ___Factor the numerator and denominator___

Step 2. _____Reduce/Cancel like terms.

Simplify

Look for common factors.

Simplify.Answer:

Under what conditions is this expression undefined?

A rational expression is undefined if the denominator equals zero. To find out when this expression is undefined, completely factor the denominator.

Answer: The values that would make the denominator equal to 0 are –7,

3, and –3. So the expression is undefined at y = –7, y = 3, and y = –3. These values are called excluded values.

a. Simplify

b. Under what conditions is this expression undefined?

Answer:

Answer: undefined for x = –5, x = 4, x = –4

Multiple-Choice Test Item

For what values of p is undefined?

A 5 B –3, 5 C 3, –5 D 5, 1, –3

Read the Test ItemYou want to determine which values of p make the denominator equal to 0.

Solve the Test ItemLook at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator.

Factor the denominator.

Solve each equation.

Answer: B

Zero Product Propertyor

Multiple-Choice Test Item

For what values of p is undefined?

A –5, –3, –2 B –5 C 5 D –5, –3

Answer: D

Simplify

Simplify.Answer: or –a

Simplify

Answer: –x

Simplify

Answer: Simplify.

Simplify

Answer: Simplify.

Simplify each expression.

a.

b.

Answer:

Answer:

Simplify

Answer: Simplify.

Simplify

Answer:

Simplify

Answer: Simplify.

Simplify

Simplify.Answer:

Stopped here after day 1

Answer: 1

Simplify each expression.

a.

b.

Answer:

Simplify

Express as adivision expression.

Multiply by thereciprocal of divisor.

Factor.

1 1 –1

1 1 1

Simplify.Answer:

Simplify

Answer:

Example 1 LCM of Monomials

Example 2 LCM of Polynomials

Example 3 Monomial Denominators

Example 4 Polynomial Denominators

Example 5 Simplify Complex Fractions

Example 6 Use a Complex Fraction to Solve a Problem

Find the LCM of 15a2bc3, 16b5c2, and 20a3c6.

Factor the firstmonomial.

Factor the secondmonomial.

Factor the thirdmonomial.

Use each factor the greatest number of times it appears as a factor and simplify.

Answer:

Find the LCM of 6x2zy3, 9x3y2z2, and 4x2z.

Answer: 36x3y3z2

Find the LCM of x3 – x2 – 2x and x2 – 4x + 4.

Factor the first polynomial.

Factor the second polynomial.

Answer: Use each factor the greatest number of times it appears as a factor.

Find the LCM of x3 + 2x2 – 3x and x2 + 6x + 9.

Answer:

Simplify

The LCD is 42a2b2. Find equivalent fractions that have this denominator.

Simplify each numerator and denominator.

Add the numerators.Answer:

Simplify

Answer:

Simplify

Factor the denominators.

The LCD is6(x – 5).

Subtract the numerators.

DistributiveProperty

Combine liketerms.

Simplify.

1

1

Simplify.Answer:

Simplify

Answer:

Simplify

The LCD of the numerator is ab. The LCD of the denominator is b.

Simplify the numerator and denominator.

Write as a divisionexpression.

Multiply by the reciprocal of the divisor.

1

1

Simplify.Answer:

Simplify

Answer:

Coordinate Geometry Find the slope of the line that

passes through and

Definition of slope

The LCD of the numerator is 3k. The LCD of the denominator is 2k.

Write as a division expression.

Simplify.

Answer: The slope is

Coordinate Geometry Find the slope of the line that

passes through and

Answer:

Example 1 Vertical Asymptotes and Point Discontinuity

Example 2 Graph with a Vertical Asymptote

Example 3 Graph with Point Discontinuity

Example 4 Use Graphs of Rational Functions

Determine the equations of any vertical asymptotes and the values of

x for any holes in the graph of

First factor the numerator and denominator of the rational expression.

Answer: The function is undefined for x = –2 and –3.

Since x = –3 is a vertical

asymptote and x = –2 is a hole in the graph.

1

1

Determine the equations of any vertical asymptotes and the values of

x for any holes in the graph of

Answer: vertical asymptote: x = –5; hole: x = –3

Answer:Graph

The function is undefined for

x = –1. Since is in its

simplest form, x = –1 is a vertical asymptote. Draw the vertical asymptote.

Make a table of values.

x f (x)

–4 1.33

–3 1.5

–2 2

0 0

1 0.5

2 0.67

3 0.75

Answer:

Plot the points and draw the graph.

As |x| increases, it appears that the

y values of the function get closer and closer to 1. The line with

the equation f (x) = 1 is a

horizontal asymptote of

the function.

Answer:

Graph

Answer:

Graph

Notice that or Therefore,

the graph of is the graph of

with a hole at

Answer:

Graph

Answer:

Transportation A train travels at one velocity V1 for a given amount of

time t1 and then another velocity V2 for a different amount of time t2. The average

velocity is given by

Let t1 be the independent variable and let V be the dependent

variable. Draw the graph if V1 = 50 miles per hour, V2 = 30 miles

per hour, and t2 = 1 hour.

Answer:

The function is

The vertical asymptote is Graph the vertical

asymptote and the function. Notice that the horizontal

asymptote is

What is the V-intercept of the graph?

Answer: The V-intercept

is 30.

What values of t1 and V are meaningful in the context of the problem?

Answer: In the problem context, time and velocity are positive values. Therefore, positive values

of t1 and V values between 30 and 60 are meaningful.

Transportation A train travels at one velocity V1 for a given amount of

time t1 and then another velocity V2 for a different amount of time t2. The average

velocity is given by

a. Let t1 be the independent variable and let V be the

dependent variable. Draw the graph if V1 = 60 miles

per hour, V2 = 30 miles per hour, and t2 = 1 hour.

Answer:

b. What is the V-intercept of the graph?

c. What values of t1 and V are

meaningful in the context of

the problem?

Answer: The V-intercept is 30.

Answer: t1 is positive and V is between 30

and 60.