Example No 2
Let's look at ((r - 4) ÷ (r2 - 5r + 6)) ÷ ((r - 3) ÷ (r2 - 6r + 9)).
First, we need to get the polynomials in simplified form, and that means factoring. Let's review again factoring. Let's look at r2 - 5r + 6. We're looking for multiples of 6 that add to -5.
-3 × -2 = 6, and -3 + -2 = -5. So r2 - 5r + 6 factors into (r - 3)(r - 2).
How about r2 - 6r + 9?
Well, -3 × -3 = 9 and -3 + -3 = -6, so r2 - 6r + 9 factors into (r - 3)(r - 3).
Our expression with simplified polynomials now looks like:
((r - 4) ÷ ((r - 3)(r - 2))) ÷ ((r - 3) ÷ ((r - 3)(r - 3)))
We aren't done yet. To divide fractions, we need to 'flip' the second fraction. So, we have:
((r - 4) ÷ ((r - 3)(r - 2))) × (((r - 3)(r - 3)) ÷ (r - 3))
The second step is canceling, or what I like to call slashing! Once we have 'slashed' all the like terms from the top and bottom, we multiply straight across, but don't multiply anything we slashed! So we're going to slash (r - 3) over (r - 3) and (r - 3) over (r - 3).
It turns out that ((r - 4) ÷ (r2 - 5r + 6)) ÷ ((r - 3) ÷ (r2 - 6r + 9)) = (r - 4) ÷ (r - 2).
Example No 3
Let's look at ((m + 5) ÷ (m - 9)) ÷ ((m2+ 10m + 25) ÷ (m2 - 81)).
First, we need to get the polynomials in simplified form. That means factoring. Factoring m2 + 10m + 25, we get (m + 5)(m + 5). Factoring m2 - 81, we get (m + 9)(m - 9). So our expression now looks like:
((m + 5) ÷ (m - 9)) ÷ (((m + 5)(m + 5)) ÷ ((m + 9)(m - 9)))
Next, flip! We flip the second fraction and change it to a multiplication. So our expression now looks like:
((m + 5) ÷ (m - 9)) × (((m + 9)(m - 9)) ÷ ((m + 5)(m + 5))).
Before we have our final answer, we cancel, or 'slash,' like terms. I love this part! We slash (m + 5) over (m + 5) and (m - 9) over (m - 9). Once we have 'slashed' all of the like terms from the top and the bottom, we multiply straight across. Don't multiply anything we slashed!
It turns out that ((m + 5) ÷ (m - 9)) ÷ ((m2 + 10m + 25) ÷ (m2 - 81)) = (m + 9) ÷ (m + 5).
Lesson Summary
Multiplication and division of rational expressions is easy once you remember these steps!
Multiplication
- Factor
- Slash
- Multiply
Division
- Factor
- Flip
- Slash
- Multiply
One final note - many of my students think they need to get a common denominator when multiplying or dividing fractions. No, no, no! Never! Only when you add and subtract, so please don't make that same mistake.
Try this mnemonic to help you remember:
- Add Subtract Common Denominators, Multiply Divide None.
- Auntie Sits Counting Diamonds, Mother Does Not.