Factor top and bottom, then cancel common factors. The rules are the ones for numeric fractions.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Factors, never terms
Only whole factors cancel. In (x + 3) ÷ (x + 5) the x values cannot be cancelled, because they are terms.
Multiplying
Multiply tops and bottoms, but factor and cancel first. It saves expanding something you will only cancel.
Dividing
Multiply by the reciprocal of the second expression. Flip and multiply, exactly as with numbers.
Adding
Adding needs a common denominator. Factor each denominator to build the least common one.
State the restrictions
Any value making an original denominator zero is excluded, even if the factor later cancels.
Factor before doing anything
Factoring reveals what cancels and what the least common denominator is. Multiplying denominators together always works and produces expressions far larger than necessary.
Cancel factors, never terms
A factor common to the whole numerator and whole denominator may be cancelled. A term from a sum may not: (x + 3)/(x + 5) does not reduce. This is the most persistent error in algebra.
Bracket the numerator when subtracting
The whole numerator is subtracted, so bracket it before distributing the minus. Subtracting only the first term is the standard slip and it is invisible in the finished line.
Note the restrictions
Every value excluded by any denominator along the way stays excluded from the final answer, even if that denominator cancelled. The simplified expression alone does not record them.
Step 2: Try It Yourself
Tap and try it out.
2/3
3/4
2/3 is less than 3/4.
Compare how much of the whole each one covers.
Step 3: Watch an Example
One step at a time.
Watch Ines Simplify a Quotient
Ines simplifies (x² − 9) ÷ (x² + 5x + 6).
- Step 1
The top is a difference of squares: (x − 3)(x + 3).
Step 4: Your Turn
Practice makes it stick.
The Restriction
Problem 1 of 2
For (x − 3) ÷ (x + 2), which value of x is excluded?
The Cancel
Problem 2 of 2
In (x² − 9) ÷ (x² + 5x + 6), which factor cancels? Enter 3 for (x + 3).
Simplify and Combine
1 of 8
For 1 ÷ (x − 5), which x is excluded?
2 of 8
For 1 ÷ (x + 4), which x is excluded?
3 of 8
Can the x values in (x + 3) ÷ (x + 5) be cancelled? 1 yes, 0 no.
4 of 8
To divide by a fraction, what do you multiply by? 1 the same, 2 the reciprocal.
5 of 8
(x² − 4) ÷ (x − 2) simplifies to x + k. What is k?
6 of 8
(x² − 25) ÷ (x + 5) simplifies to x − k. What is k?
7 of 8
Put the addition process in order.
- 1Build the least common denominator.
- 2Rewrite each fraction over it.
- 3Add the numerators and simplify.
- 4Factor every denominator.
8 of 8
(x² − 36) ÷ (x − 6) simplifies to x + k. What is k?
Step 5: Quick Check
Show what you know.
Question 1 of 2
For 1 ÷ (x − 9), which value of x is excluded?
Question 2 of 2
What may be cancelled from a fraction?
What You Learned
- 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.