Divide - Rational Expressions

Rational Expressions - Divide - How it Works - Video

Example 1

Example 1:

Here we have the expression 3 ÷ 1/4 . 

3 ÷ 1/4

3/1 * 4/1

3  * 4     /     1 * 1

12 / 1

12

Given

Flipped the second fraction

Smashed the numbers together

Multiplied

Simplified

We know the answer is 12 following the steps, but do we flip the fraction. Well, let's take a look at something everyone loves, PIZZA. If a pizza is cut into 4 sections, instead of the normal 6, 8, or 10, we will have 4 slices per pizza.

12 represents the amount of slices we have with 3 pizzas cut into fourths. So, 3 ÷ 1/4 in words is if we had 3 pizzas divided by the amount that we want to cut the pizza by, how many slices will there be? 

Example 2

Example 2:

Here we have the expression (x2 - 5x + 6) / (x2 + 10x - 11) ÷ (x2 + 2x - 15) / (x2 + 4x - 5). Since we have division, we have to flip the second fraction so we can repeat the steps for multiply rational expressions. After we flip, the best bet is to factor them and see if we can cancel any factors out so we can make it smaller.

Factor: x2 - 5x - 6

Factor: x2 + 10x - 11

Factor: x2 + 4x - 5

Factor: x2 + 2x - 15

So x2 - 5x - 6 becomes (x - 2)(x - 3).

So x2 + 10x - 11 becomes (x - 1)(x + 11).

So x2 + 4x - 5 becomes (x - 1)(x + 5).

So x2 + 2x - 15 becomes (x - 3)(x + 5).

Remember these questions for the most part will cancel so if you find one factor in the numerator, it will probably be in the denominator. So you can use that to help you to find other factors.

(x2 - 5x + 6) / (x2 + 10x - 11)      ÷      (x2 + 2x - 15) / (x2 + 4x - 5)

(x2 - 5x + 6) / (x2 + 10x - 11)      *     (x2 + 4x - 5) / (x2 + 2x - 15)

 (x - 2)(x - 3) / (x - 1)(x + 11)      *      (x - 1)(x + 5) / (x + 5)(x - 3)

 (x + 2)(x - 3)(x - 1)(x + 7)     /     (x - 5)(x - 2)(x + 7)(x - 2)

 (x - 3) * (x - 2) * (x - 1) * (x + 5)     /     (x - 3) * (x - 1) * (x + 5) * (x + 11)

̶(̶x̶ ̶-̶ ̶3̶)̶ * (x - 2) *   ̶(̶x̶ ̶-̶ ̶1̶)̶ ̶ * ̶(̶x̶ ̶+̶ ̶5̶)̶     /    ̶(̶x̶ ̶-̶ ̶3̶)̶ *   ̶(̶x̶ ̶-̶ ̶1̶)̶ ̶ * ̶(̶x̶ ̶+̶ ̶5̶)̶  * (x + 11)

 (x - 2)     /     (x + 11)

Given

Flipped the second fraction

Factored each polynomial

Smash the numerators and denominators together

Moved the factors around so common factors are closer

Canceled out the factors (x -3), (x - 2), and (x + 5) on  the top and bottom

Wrote the answer

Example 3

Example 3:

Here we have the expression (x2 + 5x - 6) / (3x3 + 6x2) ÷ (x2 + 12x - 13) / (3x2 - 15x - 42). Since we have division, we have to flip the second fraction so we can repeat the steps for multiply rational expressions. After we flip, the best bet is to factor them and see if we can cancel any factors out so we can make it smaller.

Factor: x2 + 5x - 6

Factor: 3x3 - 6x2

Factor: 3x2 - 15x - 42

Factor: x2 + 12x - 13

So x2 + 5x - 6 becomes (x - 1)(x + 6).

So 3x3 - 6x2 becomes 3x2(x + 2).

So 3x2 - 15x - 42 becomes 3(x + 2)(x - 7).

So x2 + 12x - 13 becomes (x - 1)(x + 13).

Remember these questions for the most part will cancel so if you find one factor in the numerator, it will probably be in the denominator. So you can use that to help you to find other factors.

(x2 + 5x - 6) / (3x3 - 6x2)      ÷      (x2 + 12x - 13) / (3x2 - 15x - 42)

(x2 + 5x - 6) / (3x3 - 6x2)      ÷      (3x2 - 15x - 42) / (x2 + 12x - 13)

 (x - 1)(x + 6) / [ 3x2(x + 2) ]      *      [ 3(x - 7)(x + 2) ] / (x + 13)(x - 1)

3(x - 1)(x + 6)(x - 7)(x + 2)     /    3x2(x + 2)(x + 13)(x - 1)

3 * (x - 7) * (x - 1) * (x + 2) * (x + 6)     /     3 * x2 * (x - 1) * (x + 2) * (x + 13)

̶3̶ * (x - 7) *   ̶(̶x̶ ̶-̶ ̶1̶)̶ ̶ *  ̶(̶x̶ ̶+̶ ̶2̶)̶ ̶ * (x + 6)     /     ̶3̶ * x2 *   ̶(̶x̶ ̶-̶ ̶1̶)̶ ̶ *  ̶(̶x̶ ̶+̶ ̶2̶)̶ ̶* (x + 13)

 (x - 7)(x + 6)     /     [ x2(x + 13) ]

Given

Flipped the second fraction

Factored each polynomial

Smash the numerators and denominators together

Moved the factors around so common factors are closer

Canceled out the factors (x -3), (x - 2), and (x + 5) on  the top and bottom

Wrote the answer

Live Worksheet