Multiplying and Multiplying and Dividing Rational Dividing Rational

ExpressionsExpressions

Multiplying and Multiplying and Dividing Rational Dividing Rational

ExpressionsExpressionsUnit 3Unit 3

English CasbarroEnglish Casbarro

Definition: A rational expression is a ratio of polynomialsBecause it is a ratio, it behaves like a ratio you can cancel outvariable expressions that “match”. They must be connected bymultiplication, not addition or subtraction for this to work.

Example 1

Simplify. Identify all x-values for which the expression is undefined.

You can simplify the x variable: x7-4 = x3. The final versionof the expression is:

The final, simplified version doesn’t have any x’s in the denominator, butto find the values that must be excluded, you need to look at the original. Sincethere was an x in the denominator, then x ≠ 0.

Example 2

Simplify. Identify all x-values for which the expression is undefined. In thisexpression, all of the terms are added and subtracted, so you must factor inorder to simplify. (multiplication)

The x-values that must be excluded are – 4 and –1, so your answer would be:

Recall: when we were graphing on the first day, the intro talked about aremovable discontinuity. The -1 is the removable discontinuity because it’s not obvious in the final answer that this is an excluded point. The -4 isnot removable because it is still part of the final answer, and you can see thatit must be excluded (otherwise the expression is undefined).

Multiplication and division of rational expressions follows the same rulesas multiplication and division of fractions do.

Recall:

Before you multiplied, you reduced the numbers by their factors. You will dothe same thing here. Don’t forget to write your restrictions!!

Example 3

Division is the same as multiplication, except for the “invert and multiply”step on the second fraction. Don’t forget your restrictions!

Example 4

Example 5

Turn in the following problems

1. For a car moving with initial speed v0 and acceleration a, the distance d that the car travels in time t is given by d = v0t + ½at2.a. Write a rational expression in terms of t for the average speed of the car during a period of acceleration. Simplify the expression.b. During a race, a driver accelerates for 3s at a rate of 10 ft/s2 in order to pass another car. The driver’s initial speed was 264 ft/s. What was the driver’s average speed during acceleration?

2. In an auto race, a car with an average speed of 200 mi/h takes an average of 31.5 s to complete one lap of the track.a. Write an inverse variation function that gives the average speed s of a car in miles per hour as a function of the time t in seconds needed to complete one lap.b. How many seconds does it take the car to complete one lap at an average speed of 210 mi/h?

3. For which values of x is the expression undefined?

A. 0 and 1 B. 1 and 2 C. –1 and 2 D. –2 and 1