Lesson Objectives

- Demonstrate an understanding of Rational Expressions
- Demonstrate an understanding of how to Simplify a Rational Expression
- Learn how to Multiply Rational Expressions
- Learn how to Divide Rational Expressions

## How to Multiply & Divide Rational Expressions

In our pre-algebra course, we learned how to multiply fractions.
When we multiply fractions, we multiply numerators and place the result over the product of the denominators. We usually cross cancel before we multiply. This
will allow us to report a simplified answer. Let's take a look at a quick example:
$$\frac{2}{3} \cdot \frac{3}{5}$$
Before we multiply, let's cross cancel a 3 between the two fractions:
$$\require{cancel} \frac{2}{1\cancel{3}} \cdot \frac{1\cancel{3}}{5}$$
Now we can multiply across:
$$\frac{2 \cdot 1}{1 \cdot 5} = \frac{2}{5}$$
Additionally, we also learned how to divide fractions. When we divide fractions,
we multiply the first fraction (leftmost) by the reciprocal of the second (rightmost). Let's take a look at a quick example:
$$\frac{5}{18} \div \frac{25}{27}$$
Let's set this problem up as a multiplication. The first fraction (leftmost) stays the same. We will take the reciprocal of the second fraction:
$$\frac{5}{18} \cdot \frac{27}{25}$$
We can cross cancel a 5 and a 9 between the fractions:
$$\frac{1\cancel{5}}{2\cancel{18}} \cdot \frac{3\cancel{27}}{5\cancel{25}}$$
Now we can multiply across:
$$\frac{1 \cdot 3}{2 \cdot 5} = \frac{3}{10}$$

Example 1: Find each product $$\frac{x-7}{x+2} \cdot \frac{3x + 6}{x - 7}$$ Step 1) Factor each numerator and denominator: $$\frac{x - 7}{x + 2} \cdot \frac{3(x + 2)}{x - 7}$$ Step 2) Cancel all common factors: $$\frac{1\cancel{(x - 7)}}{1\cancel{(x + 2)}} \cdot \frac{3\cancel{(x + 2)}}{1\cancel{(x - 7)}}$$ Step 3) Multiply numerators and place this result over the product of the denominators: $$\frac{1 \cdot 3}{1 \cdot 1} = 3$$ Example 2: Find each product $$\frac{1}{x - 6} \cdot \frac{x^2-8x+12}{x-8}$$ Step 1) Factor each numerator and denominator: $$\frac{1}{(x - 6)} \cdot \frac{(x-6)(x-2)}{(x-8)}$$ Step 2) Cancel all common factors: $$\frac{1}{1\cancel{(x - 6)}} \cdot \frac{\cancel{(x-6)}(x-2)}{(x-8)}$$ Step 3) Multiply numerators and place this result over the product of the denominators: $$\frac{1 \cdot (x - 2)}{1 \cdot (x - 8)} = \frac{x - 2}{x - 8}$$

Example 3: Find each quotient $$\frac{5}{30x - 18x^2} \div \frac{5x - 10}{9x - 15}$$ Step 1) Let's set up our multiplication: $$\frac{5}{30x - 18x^2} \cdot \frac{9x - 15}{5x - 10}$$ Step 2) Factor each numerator and denominator $$\frac{5}{6x(5 - 3x)} \cdot \frac{3(3x - 5)}{5(x - 2)}$$ Step 3) Cancel all common factors:

In this problem, we see an example of a common occurrence. We have opposite factors:

(5 - 3x) and (3x - 5)

When this occurs, they cancel as (-1). We can see this by factoring a -1 out of one of the factors: $$\frac{5}{-6x(3x-5)} \cdot \frac{3(3x - 5)}{5(x - 2)}$$ Notice how we have (-6x). We factored out a (-1) so that we could write our factor as: 3x - 5. Now we can cancel: $$\frac{5}{-6x\cancel{(3x-5)}} \cdot \frac{3\cancel{(3x - 5)}}{5(x - 2)}$$ This leaves us with: $$\frac{5}{-6x} \cdot \frac{3}{5(x-2)}$$ We can cross cancel a 5 and a 3 between the numerator and denominator: $$\frac{1\cancel{5}}{-2\cancel{6}x} \cdot \frac{1 \cancel{3}}{1\cancel{5}(x-2)}$$ Step 4) Multiply numerators and place this result over the product of the denominators: $$\frac{1 \cdot 1}{-2x \cdot (x - 2)} = -\frac{1}{2x(x-2)}$$

### Multiplying Rational Expressions

- Factor each numerator and denominator
- Cancel common factors between all numerators and all denominators
- Multiply numerators and place this result over the product of the denominators

Example 1: Find each product $$\frac{x-7}{x+2} \cdot \frac{3x + 6}{x - 7}$$ Step 1) Factor each numerator and denominator: $$\frac{x - 7}{x + 2} \cdot \frac{3(x + 2)}{x - 7}$$ Step 2) Cancel all common factors: $$\frac{1\cancel{(x - 7)}}{1\cancel{(x + 2)}} \cdot \frac{3\cancel{(x + 2)}}{1\cancel{(x - 7)}}$$ Step 3) Multiply numerators and place this result over the product of the denominators: $$\frac{1 \cdot 3}{1 \cdot 1} = 3$$ Example 2: Find each product $$\frac{1}{x - 6} \cdot \frac{x^2-8x+12}{x-8}$$ Step 1) Factor each numerator and denominator: $$\frac{1}{(x - 6)} \cdot \frac{(x-6)(x-2)}{(x-8)}$$ Step 2) Cancel all common factors: $$\frac{1}{1\cancel{(x - 6)}} \cdot \frac{\cancel{(x-6)}(x-2)}{(x-8)}$$ Step 3) Multiply numerators and place this result over the product of the denominators: $$\frac{1 \cdot (x - 2)}{1 \cdot (x - 8)} = \frac{x - 2}{x - 8}$$

### Dividing Rational Expressions

- Set up the multiplication of the first (leftmost) rational expression times the reciprocal of the second
- Factor each numerator and denominator
- Cancel common factors between all numerators and all denominators
- Multiply numerators and place this result over the product of the denominators

Example 3: Find each quotient $$\frac{5}{30x - 18x^2} \div \frac{5x - 10}{9x - 15}$$ Step 1) Let's set up our multiplication: $$\frac{5}{30x - 18x^2} \cdot \frac{9x - 15}{5x - 10}$$ Step 2) Factor each numerator and denominator $$\frac{5}{6x(5 - 3x)} \cdot \frac{3(3x - 5)}{5(x - 2)}$$ Step 3) Cancel all common factors:

In this problem, we see an example of a common occurrence. We have opposite factors:

(5 - 3x) and (3x - 5)

When this occurs, they cancel as (-1). We can see this by factoring a -1 out of one of the factors: $$\frac{5}{-6x(3x-5)} \cdot \frac{3(3x - 5)}{5(x - 2)}$$ Notice how we have (-6x). We factored out a (-1) so that we could write our factor as: 3x - 5. Now we can cancel: $$\frac{5}{-6x\cancel{(3x-5)}} \cdot \frac{3\cancel{(3x - 5)}}{5(x - 2)}$$ This leaves us with: $$\frac{5}{-6x} \cdot \frac{3}{5(x-2)}$$ We can cross cancel a 5 and a 3 between the numerator and denominator: $$\frac{1\cancel{5}}{-2\cancel{6}x} \cdot \frac{1 \cancel{3}}{1\cancel{5}(x-2)}$$ Step 4) Multiply numerators and place this result over the product of the denominators: $$\frac{1 \cdot 1}{-2x \cdot (x - 2)} = -\frac{1}{2x(x-2)}$$

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