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Rational Expressions & Functions

Objective: To define and simplify rational
expressions, describe their domains, and
multiply and divide rational expressions.

Definition 1

Rational expression – an algebraic
__________ with a variable

Examples

Remember: Denominator cannot equal zero!

Definition 2

rational function – a __________ of two
polynomials

where p(x) and q(x) are polynomials and
where q(x) is not zero.

Find all numbers in the domain of each
rational function. Write using interval
notation.

Fundamental Property
of Rational Numbers

See bottom of page 415.

Simplifying a Rational Expression

1. ______ numerator and denominator.
2. Apply the fundamental property.
(__________ common factors.)
3. __________.
4. Note exclusions. (values that make
denominator equal zero)

Watch & think.
Simplify. Note exclusions.

EXAMPLES – Simplify.

Opposite Factors

After factoring, if you have “__________
factors”, you can factor out –1 and then
cancel.

For example: (x - 3) & (3 - x) are
opposites.
- (3 - x) =(-3 + x)= (x – 3)

Simplify.

Multiplying and Dividing
Rational Expressions

Page 418

Review of Multiplying Fractions

Rule: top x top
bottom x bottom

Want to Cancel before Multiplying? Fine!
(It’s like reducing first!) But…
Cancel only top to bottom!

Multiplying Rational Expressions

1. __________, if possible.
2. Cancel out common __________ on top &
bottom.
3. Multiply remaining factors.
4. Simplify, if needed.

Multiplication Examples

Dividing Algebraic Fractions

Reminder: When dividing fractions,
multiply by the __________. (Flip the
second fraction and multiply.)

Steps for Dividing
Rational Expressions

1. Flip second fraction and change division sign
to multiplication sign.
2. Follow same steps as with multiplication.

Division Examples

IMPORTANT REMINDER!!!!

You can only cancel a monomial with a
monomial, or a binomial with a binomial, etc.
You CANNOT cancel part of a polynomial
with a monomial!!!
You MUST __________ before canceling!!

SUMMARY

Factor, cancel, multiply.
If dividing, multiply by reciprocal of 2^nd
fraction.

Rational Expressions & Functions

Definition 1

Definition 2

Multiplying and Dividing Rational Expressions

Multiplying and Dividing
Rational Expressions