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Simplifying Rational Expressions

Simplifying Rational Expressions

Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.

Simplified Rational Expression

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

For example:

  • \(\frac{2}{3}\) is simplified because there are no common factors of 2 and 3.
  • \(\frac{2x}{3x}\) is not simplified because x is a common factor of 2x and 3x.

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.

Equivalent Fractions Property

If a, b, and c are numbers where \(b\ne 0,c\ne 0\), then \(\frac{a}{b}=\frac{a·c}{b·c}\) and \(\frac{a·c}{b·c}=\frac{a}{b}\).

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see \(b\ne 0,c\ne 0\) clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Let’s start by reviewing how we simplify numerical fractions.

Example

Simplify: \(-\frac{36}{63}.\)

Solution

 .
Rewrite the numerator and denominator showing the common factors..
Simplify using the Equivalent Fractions Property..

Notice that the fraction \(-\frac{4}{7}\) is simplified because there are no more common factors.

Throughout this tutorial, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x\ne 0\) and \(y\ne 0\).

Example

Simplify: \(\frac{3xy}{18{x}^{2}{y}^{2}}.\)

Solution

 .
Rewrite the numerator and denominator showing the common factors..
Simplify using the Equivalent Fractions Property..

Did you notice that these are the same steps we took when we divided monomials in the tutorial on polynomials?

To simplify rational expressions we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

This figure contains three columns. The first column, shows the numerator and denominator in factored form. The numerator has 2 times 3 times 7. The denominator has 3 times 5 times 7. The common factors, 3 and 7 are crossed out. The second row, first column shows what remains after the threes and sevens are crossed out, which is 2 over 5 in fraction form. The last row in the first column reads “We removed the common factors of 3 and 7. They are the factors of the product.” The first row of the middle column shows 3 x and then x minus 9 in parentheses in the numerator. The denominator shows 5 and then x-9 in parentheses. The common factors x minus 9 are crossed out. The second row of the middle column shows what remains after removing the common factors, which is 3 x over 5 in fraction form. The last row in the middle column reads, “We removed the common factor x minus 9. It is a factor of the product.” The first row of the third column shows x plus 5 in the numerator and x in the denominator. The second row says “No common factors” and the third row reads, “While there is an x in both the numerator and the denominator, the x in the numerator is a term of a sum”.

Note that removing the x’s from \(\frac{x+5}{x}\) would be like cancelling the 2’s in the fraction \(\frac{2+5}{2}\)!

Example: How to Simplify Rational Binomials

Simplify: \(\frac{2x+8}{5x+20}.\)

Solution

This figure is a table with three columns and two rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell says “Step 1. Factor the numerator and denominator completely.” The second cell says “Factor 2x plus 8 and 5x minus 20.” The third cell contains 2x plus 8, divided by 5x plus 20. Below this is 2 times x plus 4 divided by 5 times x plus 4.In the second row, the first cell says “Step 2. Simplify by dividing out common factors.” The second cell says “Divide out the common factors.” The third cell contains 2 times x plus 4 divided by 5 times x plus 4, where x plus 4 cancels out in the numerator and the denominator. It simplifies to 2 fifths.

We now summarize the steps you should follow to simplify rational expressions.

Simplify a Rational Expression.

  1. Factor the numerator and denominator completely.
  2. Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors!

We’ll use the methods we covered in the tutorial on factoring to factor the polynomials in the numerators and denominators in the following examples.

Example

Simplify: \(\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}\hfill \\ \text{Factor the numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+6\right)}\hfill \\ \\ \begin{array}{c}\text{Remove the common factor}\phantom{\rule{0.2em}{0ex}}x+2\phantom{\rule{0.2em}{0ex}}\text{from}\hfill \\ \text{the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(x+2\right)}\left(x+3\right)}{\require{cancel}\cancel{\left(x+2\right)}\left(x+6\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{x+3}{x+6}\hfill \end{array}\)

Can you tell which values of x must be excluded in this example?

Example

Simplify: \(\frac{{y}^{2}+y-42}{{y}^{2}-36}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}+y-42}{{y}^{2}-36}\hfill \\ \text{Factor the numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y+7\right)\left(y-6\right)}{\left(y+6\right)\left(y-6\right)}\hfill \\ \\ \begin{array}{c}\text{Remove the common factor}\phantom{\rule{0.2em}{0ex}}y-6\phantom{\rule{0.2em}{0ex}}\text{from}\hfill \\ \text{the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y+7\right)\require{cancel}\cancel{\left(y-6\right)}}{\left(y+6\right)\require{cancel}\cancel{\left(y-6\right)}}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{y+7}{y+6}\hfill \end{array}\)

Example

Simplify: \(\frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10}\hfill \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{using grouping to factor the numerator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{2}\left(p-2\right)+2\left(p-2\right)}{\left(p-5\right)\left(p-2\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left({p}^{2}+2\right)\left(p-2\right)}{\left(p-5\right)\left(p-2\right)}\hfill \\ \\ \begin{array}{c}\text{Remove the common factor of}\phantom{\rule{0.2em}{0ex}}p-2\hfill \\ \text{from the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left({p}^{2}+2\right)\require{cancel}\cancel{\left(p-2\right)}}{\left(p-5\right)\require{cancel}\cancel{\left(p-2\right)}}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{2}+2}{p-5}\hfill \end{array}\)

Example

Simplify: \(\frac{2{n}^{2}-14n}{4{n}^{2}-16n-48}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{2{n}^{2}-14n}{4{n}^{2}-16n-48}\hfill \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{2n\left(n-7\right)}{4\left({n}^{2}-4n-12\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{2n\left(n-7\right)}{4\left(n-6\right)\left(n+2\right)}\hfill \\ \\ \text{Remove the common factor, 2.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{2}n\left(n-7\right)}{\require{cancel}\cancel{2}·2\left(n-6\right)\left(n+2\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n\left(n-7\right)}{2\left(n-6\right)\left(n+2\right)}\hfill \end{array}\)

Example

Simplify: \(\frac{3{b}^{2}-12b+12}{6{b}^{2}-24}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{3{b}^{2}-12b+12}{6{b}^{2}-24}\hfill \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3\left({b}^{2}-4b+4\right)}{6\left({b}^{2}-4\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3\left(b-2\right)\left(b-2\right)}{6\left(b+2\right)\left(b-2\right)}\hfill \\ \\ \text{Remove the common factors of}\phantom{\rule{0.2em}{0ex}}b-2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3.\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{3}\left(b-2\right)\require{cancel}\cancel{\left(b-2\right)}}{\require{cancel}\cancel{3}·2\left(b+2\right)\require{cancel}\cancel{\left(b-2\right)}}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{b-2}{2\left(b+2\right)}\hfill \end{array}\)

Example

Simplify: \(\frac{{m}^{3}+8}{{m}^{2}-4}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{m}^{3}+8}{{m}^{2}-4}\hfill \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{using the formulas for sum of cubes and}\hfill \\ \text{difference of squares.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(m+2\right)\left({m}^{2}-2m+4\right)}{\left(m+2\right)\left(m-2\right)}\hfill \\ \text{Remove the common factor of}\phantom{\rule{0.2em}{0ex}}m+2.\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(m+2\right)}\left({m}^{2}-2m+4\right)}{\require{cancel}\cancel{\left(m+2\right)}\left(m-2\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{m}^{2}-2m+4}{m-2}\hfill \end{array}\)

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