Simplifying rational expressions

Here you will learn about simplifying rational expressions (algebraic fractions), including different powers of x, quadratics, and the difference of two squares.

Students first learn how to work with rational expressions in Algebra I and expand that knowledge as they deeply explore rational functions in Algebra II.

What is simplifying rational expressions?

Simplifying rational expressions is simplifying an algebraic fraction into its lowest terms. Similar to how you simplified numerical fractions, you need to find the greatest common factor of the numerator and denominator of the rational expression to cancel out.

When you simplified a numerical fraction like, \cfrac{6}{24}, you looked for the greatest common factor of the numerator and the denominator.

To find the greatest common factor (GCF) , you can break the numerator and the denominator down into its prime factors.

The prime factors they have in common is the greatest common factor.

The greatest common factor is: 2\times{3}=6

Now let’s look at a rational expression to simplify.

To simplify this rational expression, use the same strategy of finding the greatest common factor to cancel out of the numerator and the denominator.

To find the greatest common factor of both the numerator and denominator, break them down into prime factors.

The greatest common factor is the factors common to both the numerator and the denominator.

The greatest common factor of the numerator and denominator is a \times b=a b.

The simplified rational expression is, \cfrac{a}{b^2}.

Let’s look at another example which has quadratic expressions in the numerator and denominator of the rational expression.

Before finding the greatest common factor, you have to factor.

To find the greatest common factor, look for the binomial factors that match to cancel out.

The simplified rational expression is, \cfrac{x-3}{x+5}.

What is simplifying rational expressions?

Common Core State Standards

How does this apply to high school math?

• High School Algebra: Arithmetic with Polynomials and Rational Expressions (HSA-APR.D.6)
  Rewrite simple rational expressions in different forms; write \cfrac{a(x)}{b(x)} in the form q(x)+\cfrac{r(x)}{b(x)}, where a(x), \, b(x), \, q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

How to simplify rational expressions

In order to simplify rational expressions:

1. Break the numerator and denominator into prime factors to find the \textbf{GCF}.
2. Cancel the \textbf{GCF} from the numerator and denominator.
3. Write the simplified rational expression.

Simplifying rational expression examples

Example 1: algebraic fractions

Simplify the algebraic expression,

\cfrac{x^2 y^3}{x y^2} .

1. Break the numerator and denominator into prime factors to find the \textbf{GCF}.

The greatest common factor is x \times y \times y=x y^2

2Cancel the \textbf{GCF} from the numerator and denominator.

3Write the simplified rational expression.

The simplified expression is: xy

Example 2: algebraic fractions

Simplify the algebraic expression,

\cfrac{4 a b^2}{18 a^2 b^3} .

Break the numerator and denominator into prime factors.

The greatest common factor is: 2 \times a \times b \times b=2 a b^2

Cancel the \textbf{GCF} from the numerator and denominator.

Example 3: factor GCF

Simplify the rational expression,

\cfrac{6x+12}{10x+20} .

Break the numerator and denominator into prime factors to find the \textbf{GCF}.

In this case, in order to break both the denominator and numerator down into prime factors, you need to apply a factoring strategy which is to factor out the GCF from the numerator and the GCF from the denominator.

\begin {aligned} &\text{Numerator}\text{:} \, 6 x+12=6(x+2) \rightarrow 2 \times 3 \times(x+2) \\\\ &\text{Denominator}\text{:} \, 10 x+20=10(x+2) \rightarrow 2 \times 5 \times(x+2) \end {aligned}

The greatest common factor is: 2 \times(x+2)=2(x+2)

Cancel the \textbf{GCF} from the numerator and denominator.

Example 4: difference of squares

Simplify the rational expression,

\cfrac{x-4}{x^2-16} .

Break the numerator and denominator into prime factors to find the \textbf{GCF}.

In this case, you need to factor the denominator using the difference of squares. The numerator cannot be factored.

The greatest common factor is the binomial, (x-4) .

Cancel the \textbf{GCF} from the numerator and denominator.

Example 5: quadratic ÷ quadratic

Simplify the rational function,

\cfrac{x^{2}-3x-10}{x^{2}+2x-35} .

Break the numerator and denominator into prime factors to find the \textbf{GCF}.

In this case, you need to factor the trinomial in the numerator and the denominator.

The greatest common factor is x-5 .

Cancel the \textbf{GCF} from the numerator and denominator.

Example 6:

Simplify the rational expression,

\cfrac{x^2-36}{x^2-9 x+18} .

Break the numerator and denominator into prime factors to find the \textbf{GCF}.

In this case, factor the numerator and the denominator before finding the greatest common factor.

The greatest common factor is x-6 .

Cancel the \textbf{GCF} from the numerator and denominator.

Teaching tips for simplifying rational expressions

• Make connections between simplifying numerical fractions and simplifying rational functions so that students can make connections.

• Use active learning activities such as scavenger hunts or game playing to have students review skills instead of giving them a worksheet.

• Encourage students who are struggling to use digital platforms such as Khan Academy so they can view tutorial videos.

Easy mistakes to make

• Forgetting to factor expressions
  For example when simplifying the rational expression,
  \cfrac{5 x+20}{10 x+60}, cancelling out terms before factoring
  
  
  \cfrac{5 x+20}{10 x+60} ≠ \cfrac{1}{2 x+3}
  
  Instead, factor the numerator and the denominator using a factoring strategy first, then look to cancel out the greatest common factor.
  
  
  \cfrac{5 x+20}{10 x+60}=\cfrac{x+4}{2(x+6)}

• Placing a \bf{0} in the numerator or denominator when everything cancels
  For example, when simplifying the rational expression \cfrac{3 x}{9 x^2}, \, 3x is the greatest common factor, \cfrac{3 x}{9 x^2}=\cfrac{3 x}{3 \times 3 \times x \times x} ≠ \cfrac{0}{3 x}
  
  
  Instead, when all the terms cancel, you are left with a 1.
  
  \cfrac{3 x}{9 x^2}=\cfrac{3 x}{3 \times 3 \times x \times x}=\cfrac{1}{3 x}

• Factoring incorrectly
  For example, when factoring a binomial, thinking that you just put in parenthesis, 5 x+10 ≠ 5(x+10). You have to divide the factor you are factoring out by all the terms in the parenthesis, 5 x+10=5(x+2) .
  
  Factoring a trinomial incorrectly by not being mindful of the signs. For example, when factoring x^2-10 x+24 ≠ (x+4)(x+6)
  
  Be careful of the signs, and notice the - \, 10 as the coefficient of the b -term.
  
  x^2-10 x+24=(x-4)(x-6)
  
  For more help with factoring go to:
  
  Step-by-step guide: Factoring

Related rational function lessons

• Rational functions
• Adding and subtracting rational expressions
• Multiplying rational expressions

The next lessons are

• Vectors

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