Operations With Rational Expressions

Fractions, with polynomials inside.

12 problems · about 21 minutes · A-APR.D.7

What this lesson teaches

The student simplifies, multiplies, divides, adds and subtracts rational expressions.

Factor first, then cancel whole factors only.
Divide by multiplying by the reciprocal.
Adding needs a common denominator, and restrictions come from the original denominators.

Straightforward practice. Get the method working first.

5 problems

For 1 ÷ (x − 9), which value of x is excluded?
Answer 9
Why 9.
What may be cancelled from a fraction?
Answer Whole factors only.
Why Only factors.
For 1 ÷ (x − 5), which x is excluded?
Answer 5
Why x − 5 = 0.
For 1 ÷ (x + 4), which x is excluded?
Answer -4
Why x + 4 = 0.
Can the x values in (x + 3) ÷ (x + 5) be cancelled? 1 yes, 0 no.
Answer 0
Why They are terms, not factors.

The same ideas with more to keep track of.

3 problems

To divide by a fraction, what do you multiply by? 1 the same, 2 the reciprocal.
Answer 2
Why Flip and multiply.
(x² − 4) ÷ (x − 2) simplifies to x + k. What is k?
Answer 2
Why Difference of squares.
(x² − 25) ÷ (x + 5) simplifies to x − k. What is k?
Answer 5
Why Difference of squares.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the addition process in order.
Answer 1. Factor every denominator. 2. Build the least common denominator. 3. Rewrite each fraction over it. 4. Add the numerators and simplify.
Why Factoring comes before the common denominator.
(x² − 36) ÷ (x − 6) simplifies to x + k. What is k?
Answer 6
Why Difference of squares.
The Restriction: For (x − 3) ÷ (x + 2), which value of x is excluded?
Answer -2
Why −2.
The Cancel: In (x² − 9) ÷ (x² + 5x + 6), which factor cancels? Enter 3 for (x + 3).
Answer 3
Why (x + 3).