Chapter 9: Rational Expressions and Equations

-Basically we are looking at expressions and equations where there is a variable in a denominator

9.1 Multiplying and Dividing Rational Expressions

• Definitions and issues• Simplifying• Multiplying• Dividing• Complex Fractions

Definition• A rational expression is a ratio of two polynomial expressions• For example, (8 + x) / (13 + x)• Generally, we can simplify rational expressions by cancelling

out any factors common to the numerator and denominator• Note that in the expression above, you CANNOT cancel the x

terms.. You are NOT allowed to cancel terms that are WITHIN a sum or a difference

• To see why, suppose we had (3 + 5) / (3 + 8)• This is equivalent to 8 / 11… BUT if we cancelled the 3’s, we

would obtain 5 /8.. Which does NOT equal 8/11!!!• TO SIMPLIFY A RATIONAL EXPRESSION, FACTOR THE

NUMERATOR AND THE DENOMINATOR… THEN CANCEL ANY COMMON FACTORS

Issues• Before you simplify a rational expression or combine rational

expressions, you must look at the the denominator(s) and note any values which, when substituted in for a variable, would cause that denominator to equal zero

• These values are called EXCLUDED values• You must find excluded values BEFORE you begin simplifying..

And list them along with your answer• You may need to FACTOR a denominator to determine the

excluded values… however, we usually need to factor the denominator ANYHOW

• See the examples on the next few slides• Sometimes we ignore excluded values if there are multiple

variables in the rational expression

Simplify

Look for common factors.

1

1

Factor.

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

Factor the numeratorand the denominator.

Simplify.Answer: or –a

or

1

1

a

1

Simplify

Answer: –x

Multiplying two rational expressions

• Factor the numerator AND denominator of each rational expression

• List the excluded values• Cancel any factors common to the numerator and

denominator• Multiply the remaining factors in the numerator• Multiply the remaining factors in the denominator• One trick: sometimes it is advantage to factor a

negative one (-1) from an expression, if it will allow you to cancel another factor out

Simplify

Simplify.

Answer: Simplify.

Factor.

1 1 1 1 1 1 1

1 1 1 1 1 11

Simplify

Factor.

1 1 1 1 1 1 1

1 1 1 1 1 11

1

Answer: Simplify.

Simplify each expression.

a.

b.

Answer:

Answer:

Dividing Rational Expressions

• Recall that dividing by a fraction is the same as multiplying by the recipricol of that fraction

• Generally, it is advisable to rewrite a division problem as a multiplication problem before factoring and cancelling, etc.

Simplify

Answer: Simplify.

Simplify.

Factor.

1 1 1 1 1 1 1

1 1 1 1 1 11

Multiply by the reciprocal of divisor.

Simplify

Answer:

Simplify

Multiply bythe reciprocalof the divisor.

1 –1 1

1 1 1

Answer: Simplify.

Simplify

Multiply by thereciprocal of the divisor.

Simplify.Answer:

Factor.

1

1 1

1

Answer: 1

Simplify each expression.

a.

b. Answer:

COMPLEX FRACTIONS

• A complex fraction is a rational expression whose numerator and/or denominator CONTAINS another rational expression!

• It’s kind of like a fraction within a fraction• Just remember to treat this problem as a

division problem – the numerator is being divided by the denominator

Simplify

Express as adivision expression.

Multiply by thereciprocal of divisor.

Factor.

1 1 –1

1 1 1

Simplify.Answer:

Simplify

Answer: