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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 Divide Rational Expressions

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

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Page 1: 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

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

Page 2: 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

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.

Page 3: 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

Look for common factors.

Simplify.Answer:

Page 4: 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

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.

Page 5: 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

a. Simplify

b. Under what conditions is this expression undefined?

Answer:

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

Page 6: 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

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.

Page 7: 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

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

Page 8: 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

Multiple-Choice Test Item

For what values of p is undefined?

A –5, –3, –2 B –5 C 5 D –5, –3

Answer: D

Page 9: 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

Simplify.Answer: or –a

Page 10: 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

Answer: –x

Page 11: 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

Answer: Simplify.

Page 12: 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

Answer: Simplify.

Page 13: 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 each expression.

a.

b.

Answer:

Answer:

Page 14: 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

Answer: Simplify.

Page 15: 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

Answer:

Page 16: 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

Answer: Simplify.

Page 17: 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

Simplify.Answer:

Page 18: 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

Stopped here after day 1

Page 19: 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

Answer: 1

Simplify each expression.

a.

b.

Answer:

Page 20: 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

Express as adivision expression.

Multiply by thereciprocal of divisor.

Page 21: 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

Factor.

1 1 –1

1 1 1

Simplify.Answer:

Page 22: 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

Answer:

Page 23: 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
Page 24: 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

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

Page 25: 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

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

Factor the firstmonomial.

Factor the secondmonomial.

Factor the thirdmonomial.

Page 26: 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

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

Answer:

Page 27: 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

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

Answer: 36x3y3z2

Page 28: 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

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.

Page 29: 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

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

Answer:

Page 30: 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

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

Simplify each numerator and denominator.

Add the numerators.Answer:

Page 31: 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

Answer:

Page 32: 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

Factor the denominators.

The LCD is6(x – 5).

Subtract the numerators.

Page 33: 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

DistributiveProperty

Combine liketerms.

Simplify.

1

1

Simplify.Answer:

Page 34: 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

Answer:

Page 35: 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

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

Page 36: 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 the numerator and denominator.

Write as a divisionexpression.

Multiply by the reciprocal of the divisor.

1

1

Page 37: 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.Answer:

Page 38: 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

Answer:

Page 39: 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

Coordinate Geometry Find the slope of the line that

passes through and

Definition of slope

Page 40: 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

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

Write as a division expression.

Simplify.

Answer: The slope is

Page 41: 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

Coordinate Geometry Find the slope of the line that

passes through and

Answer:

Page 42: 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
Page 43: 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
Page 44: 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

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

Page 45: 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

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.

Page 46: 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

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

Page 47: 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

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

Page 48: 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

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.

Page 49: 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

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.

Page 50: 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

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:

Page 51: 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

Graph

Answer:

Page 52: 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

Graph

Notice that or Therefore,

the graph of is the graph of

with a hole at

Page 53: 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

Answer:

Page 54: 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

Graph

Answer:

Page 55: 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

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.

Page 56: 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

Answer:

The function is

The vertical asymptote is Graph the vertical

asymptote and the function. Notice that the horizontal

asymptote is

Page 57: 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

What is the V-intercept of the graph?

Answer: The V-intercept

is 30.

Page 58: 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

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.

Page 59: 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

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.

Page 60: 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

Answer:

Page 61: 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

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.


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