Multiplying And Dividing Rational Expressions

Why is this concept useful?: Multiplying and dividing rational expressions is useful because it helps to combine multiple expressions into a simplified version. Once they are simplified, it will help with solving more difficult problems.
Where does this concept fit into the curriculum?: Algebra 1

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What is a rational expression? A rational expression is a fraction made up of polynomials. Both the numerator and the denominator will be a polynomial.

How to multiply and divide rational expressions?

Multiplying Rational Expressions:

When multiplying rational expressions there are 3 main steps:

Multiply the numerators.
Multiply the denominators.
Rewrite as a rational expression and simplify when possible.

Let’s look at an example:

In the example, below, we will first multiply the numerators

The numerators are 5x and 2x.

(5x)(2x)= 10x².

Next multiply the denominators.

The denominators are 3 and 7.

3x7=21.

Now rewrite this as a rational expression.

Our product will be:

\frac{10x^2}{21}

This can not be simplified any further, so this is the final answer.

Dividing Rational Expressions:

When dividing rational expressions there are 4 main steps:

Multiply the first expression times the reciprocal of the second expression (Just like dividing fractions).
Multiply the numerators.
Multiply the denominators.
Rewrite as a rational expression and simplify when possible.

Let’s look at an example:

In the example, below, we will first multiply the first expression times the reciprocal of the second expression:

\frac{5}{(x+3)} \div (x+3)

When we rewrite with the reciprocal it becomes:

\frac{5}{x+3} x\frac{1}{x+3}

Now we need to multiply the numerators. The numerators are 5 and 1.

(5)(1)= 5

Next multiply the denominators.

The denominators are (x+3) and (x+3).

(x+3)(x+3)= x²+3x+3x+9= x²+6x+9

Now rewrite this as a rational expression.

Our quotient will be:

\frac{5}{x^2+6x+9}

This can not be simplified any further.

Sample Math Problems

Question

Find the product.

(\frac{2}{7x^2} ) (\frac{3}{8x})

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Practice Math Problems

1.) Multiply.

( - \frac{7}{6x^5} )( - \frac{6}{8x^3} )

2.) Multiply.

( \frac{2x+1}{7} )( \frac{5x+1}{7} )

3.) Divide.

( 10-2x )\div \frac{25x-5x^2}{10}

4.) Divide.

\frac{x}{(5x+2)} \div \frac{1}{(5x+4)}

Solution

First, multiply the numerators.

(2)(3)=6

Now multiply the denominators.

(7x²)(8x)= 56x³

Now rewrite this as a rational expression.

Our product will be:

\frac{6}{56x^3}

Now simplify. In this problem both the 6 and 56 can be divided by 2.

The final product is:

\frac{3}{28x^3}

Question

Find the product.

(\frac{6x^2}{9}) (-\frac{2x^2}{7})

Solution

First, multiply the numerators.

(6x²)(-2x²)= -12x^4

Now multiply the denominators.

9 x 7= 63

Now rewrite this as a rational expression.

Our product will be:

- \frac{12x^4}{63}

Now simplify. In this problem both the 12 and 63 can be divided by 3.

The final product is:

\frac{-4x^4}{21}

Question

Find the Quotient

Solution

\frac{7}{(x+2)} \div 5\frac{(x-2)}{(x+2)}

Multiply by the reciprocal of the second expression:

\frac{7}{x+2} * \frac{x+2}{5(x-2)}

Here the (x+2) and (x+2) cancel each other out.

\frac{7}{1} * \frac{1}{5(x-2)}

The numerators are 5 and 1.

Now we need to multiply the numerators. The numerators are 7 and 1.

(7)(1)= 7

Next multiply the denominators.

The denominators are (1) and 5(x-2).

(1)5(x-2)= 5(x-2)

Now rewrite this as a rational expression.

Our quotient will be:

\frac{7}{5(x-2)}

This can not be simplified any further.

Question

Find the quotient

\frac{x}{3x-1} \div (x-2)

Solution

Multiply by the reciprocal of the second expression:

\frac{x}{3x-1} * \frac{1}{x-2}

Now we need to multiply the numerators. The numerators are x and 1.

(x)(1)= x

Next multiply the denominators.

The denominators are (3x-1) and (x-2).

(3x-1)(x-2)= 3x²-6x-x+2= 3x²-7x+2

Now rewrite this as a rational expression.

Our quotient will be:

\frac{x}{3x^2-7x+2}

This can not be simplified any further.

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