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(Video) Divide Rational Expressions

by mathman1024

Here's a video by mathman1024 showing you how to divide two rational expressions. Dividing rational expressions is a lot like multiplying them. There's just one extra step. Instead of dividing, you can flip the second fraction and multiply instead. Flipping the fraction is called using the reciprocal.

$\large{\text{Summary}}$

The example he solves is: $$\dfrac{2x^2+7x+6}{x^2-8x-20}\div\dfrac{6x^2+19x+15}{x^2-5x-50}$$

The first step is to change the division into multiplication by flipping the second fraction (i.e. taking the reciprocal). $$\begin{align} &\dfrac{2x^2+7x+6}{x^2-8x-20}\div\dfrac{\color{#8925AE}{6x^2+19x+15}}{\color{Blue}{x^2-5x-50}} =\dfrac{2x^2+7x+6}{x^2-8x-20}\cdot\dfrac{\color{Blue}{x^2-5x-50}}{\color{#8925AE}{6x^2+19x+15}} \end{align}$$

Next, factor each of the four polynomials. This will help us see what common factors we can cancel out. $$=\frac{\color{green}{(2x+3)}\color{#8925AE}{(x+2)}}{\color{#8925AE}{(x+2)}\color{Blue}{(x-10)}}\cdot\frac{(x+5)\color{Blue}{(x-10)}}{\color{green}{(2x+3)}(3x+5)}$$

As you can see, there's a factor of $(2x+3)$, a factor of $(x+2)$, and a factor of $(x-10)$ in both the numerator and denominator. Removing these, we get the final answer of: $$=\boxed{\frac{x+5}{3x+5}}$$

Expressions with Rational Numbers

A rational number is a number that can be expressed in the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \ne 0$. These are numerical fractions.

A rational expression is a fraction expressed in the form $\frac{a}{b}$, but instead of $a$ and $b$ being integers, $a$ and $b$ will be polynomials.

Some examples of expressions with rational numbers are: $$\frac{3}{x+1},\quad \frac{x-6}{x^3-1},\quad \frac{4x^2-x+1}{3}$$

Working with variables in expressions that include fractions or decimals is similar to working with regular expressions.

The main difference is that you may see more fractions.

Also, the variables might appear on the top or the bottom of a fraction.

Even though the expressions may look more complicated, the basic rules of algebra still apply.

Example 1

Let’s take a look at an example.

Simplify $\frac{4}{3x} + \frac{8+x}{x}$.

First, notice that the denominators are different. Since the denominators are different, we cannot add the two expressions. We need to multiply the second expression by $\frac{3}{3}$. $$\frac{4}{3x}+ \left(\frac{3}{3} \cdot \frac{8+x}{x}\right)$$

Multiply. $$\frac{4}{3x} + \frac{3(8+x)}{3x}$$

Simplify the expression. $$\frac{4}{3x} + \frac{24+3x}{3x}$$

Since the expressions now have the same denominator, we can add them. Just add the top numbers (numerators) and keep the bottom the same. $$\frac{4}{3x} + \frac{24+3x}{3x} = \frac{4+24+3x}{3x} = \frac{28+3x}{3x}$$

Example 2

Let’s take a look at another example.

Simplify $\frac{7}{x-3} \div \frac{19}{x-3}$ if $x \ne 3$.

Notice that we can turn the problem into a multiplication problem by taking the reciprocal of the second fraction. $$\frac{7}{x-3} \div \frac{19}{x-3} = \frac{7}{x-3} \times \frac{x-3}{19}$$

Cancel out any terms that appear in both the numerator and denominator. $$\frac{7}{x-3} \times \frac{x-3}{19} = \frac{7}{\cancel{x-3}}\times \frac{\cancel{x-3}}{19} = \frac{7}{19}$$

Now, let's look at some types of problems you might see when working with expressions that have rational numbers.

$$\begin{align} \frac{x^2-4x-45}{x^2-3x+2} \cdot \frac{x^x-7x+10}{x+5}&= \frac{(x+5)(x-9)}{(x-2)(x-1)} \cdot \frac{(x-2)(x-5)}{x+5} \\ &= \frac{\cancel{(x+5)}(x-9)}{\cancel{(x-2)}(x-1)} \cdot \frac{\cancel{(x-2)}(x-5)}{\cancel{x+5}}\\ &= \frac{(x-9)(x-5)}{x-1} \\ &=\frac{x^2-14x+45}{x-1} \\ \end{align}$$

It's always helpful to first factor the numerators and denominators. That makes the fractions easier to work with.

First, notice that the denominators are different. Since the denominators are different, we cannot subtract the three expressions. We need to multiply the second expression by $\frac{4}{4}$ and the third expression by $\frac{4x}{4x}$. $$\frac{41}{4x} - \left(\frac{8}{x} \cdot \frac{4}{4}\right) -\left(2\cdot \frac{4x}{4x}\right)$$ Multiply. $$\frac{41}{4x} - \frac{32}{4x} -\frac{8x}{4x}$$ Since the expressions now have the same denominator, we can subtract the expressions. Remember to only subtract the numerators. $$\frac{41}{4x} - \frac{32}{4x} -\frac{8x}{4x} = \frac{41-32-8x}{4x}= \frac{9-8x}{4x}$$

Intro to Dividing Rational Expressions

A rational expression is a fraction where the numerator and/or the denominator are polynomials.

When we divide rational expressions, we follow a rule that is similar to dividing regular fractions. Instead of dividing, you multiply by the reciprocal.

The reciprocal of a fraction means flipping the numerator and denominator. So dividing rational expressions turns into multiplication.

As we can see in the above image, rewrite the division as multiplication by flipping the second fraction. Next, look for factors you can simplify or cancel before multiplying. Canceling factors makes the expression easier to work with.

After simplifying, multiply the numerators together and the denominators together.

Always check that no values make the denominator equal to zero, because those values are not allowed in rational expressions.

Looking Further

Let's continue to build on this idea. In these lessons, you will learn how to:

• Add and subtract rational expressions
• Multiply rational expressions

Examples: Dividing rational expressions $\\$

Diving a fraction is equivalent to multiplying its reciprocal. Recall how we divide a fraction:

Once the division becomes multiplication, follow the steps of multiplying rational expression.

Here is an example of dividing rational expression

Try the problem below!