chapter 9: rational expressions and equations -basically we are looking at expressions and equations...
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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: