# Multiplying Rational Expressions

As with fractions, the same rule we'll apply to multiplying rational expressions. If you want to first look at operations on fractions, I recommend these two lessons:

In this lesson, we will learn how to multiply two rational expressions. Very simple steps, and all you will need are some basic concepts such as simplifying a rational expression and finding the common factor of two expressions. Operations on rational expressions require some simplification and domain finding skills as detailed in the following two lessons:

The basic rule for multiplying rational expressions is the same as the basic rule for fractions: the product of their numerators divided by the product of their denominators.

### Note:

Let four rational expressions whereas and .

Here is a step-by-step procedure for multiplying rational expressions:

• Factor all numerators and denominators;

• Cancel any common factor that appears in both a numerator and a denominator;

• Multiply numerator by numerator and denominator by denominator;

### Info!

After completing the simplification, be sure to keep the same domain for the original expression so that the entire answer is true.

## Examples of How to Multiply Rational Expressions

#### Example1:

Multiply the rational expressions and give answer in simplest form:

#### Solution:

Step 1: Always try to factor and find all the common factors in the expressions in the numerators and denominators.

The domain of this rational expression is "all real numbers except 1 and 7". i.e.

Step 2: Once the expression is factored, you can see all the common factors that appears in the numerators anddenominators.

Cancel them out, one from a numerator with one from a denominator (with same color).

Step 3: Multiply the obtained expression.

After canceling the common factors we got an expression that, logically, is not equal to the expression at the beginning, but why? In the step 1, we saw that which wasn't true of the expression . So the final answer should exclude the two values of .

#### Example2:

Multiply the rational expressions and give answer in simplest form:

#### Solution:

Step 1: Factor all expressions in the numerators and denominators.

The domain of this rational expression is "all real numbers except -3 and 4". i.e.

Step 2: Now you can see all the common factors that appears more than once in the both top and bottom.

Cancel them out, one from a numerator with one from a denominator (with same color).

Step 3: We obtained a product of two fractions,

#### Example3:

Multiply the rational expressions and give answer in simplest form:

#### Solution:

Step 1: Factor all numerators and denominators. Note that the second denominator is a quadratic trinomial you can factor it using FOIL method or Simple case method.

The domain of this rational expression is "all real numbers except 0, 1 and 3". i.e.

Step 2: Now you can see all the common factors that appears more than once in the both top and bottom.

Cancel them out, one from a numerator with one from a denominator (with same color).

Step 3: The obtained expression is already in simplest form.

#### Example4:

Multiply the rational expressions and give answer in simplest form:

#### Solution:

Step 1: Factor all numerators and denominators completly

The domain of this rational expression is "all real numbers except -3, 0, 1 and 5". i.e.

Step 2: Once the expression is factored, you can see all the common factors that appears in the numerators anddenominators.

Cancel them out, one from a numerator with one from a denominator (with same color).

Step 3: Multiply numerator by numerator and denominator by denominator of the obtained expression.

#### Example4:

Simplify the following rational-expression multiplication

#### Solution:

Note that the domain is "all real numbers except 0"

Find all the common factors of denominators and numerators, then cancel them.

After canceling the common factors, now we can multiply numerators by numerators and denominators by denominators as follow: