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Study Guides > Intermediate Algebra

Read: Add and Subtract Rational Expressions

Learning Objectives

  • Identify the domain of a sum or difference of rational expressions
  • Identify the least dcommon denominator of two rational expressions
  • Add and subtract rational expressions using a greatest common denominator
In beginning math, students usually learn how to add and subtract whole numbers before they are taught multiplication and division. However, with fractions and rational expressions, multiplication and division are sometimes taught first because these operations are easier to perform than addition and subtraction. Addition and subtraction of rational expressions are not as easy to perform as multiplication because, as with numeric fractions, the process involves finding common denominators.
Add and Subtract Add and Subtract

Adding Rational Expressions

To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, given the rational expressions

6(x+3)(x+4), and 9x(x+4)(x+5)\large\frac{6}{\left(x+3\right)\left(x+4\right)},\text{ and }\frac{9x}{\left(x+4\right)\left(x+5\right)}

The LCD would be (x+3)(x+4)(x+5)\left(x+3\right)\left(x+4\right)\left(x+5\right).

To find the LCD, we count the greatest number of times a factor appears  in each denominator, and make sure it is represented in the LCD that many times. For example, in 6(x+3)(x+4)\large\frac{6}{\left(x+3\right)\left(x+4\right)}, (x+3)\left(x+3\right) is represented once and  (x+4)\left(x+4\right) is represented once, so they both appear exactly once in the LCD. In 9x(x+4)(x+5)\large\frac{9x}{\left(x+4\right)\left(x+5\right)}, (x+4)\left(x+4\right) appears once, and (x+5)\left(x+5\right) appears once. We have already accounted for (x+4)\left(x+4\right), so the LCD just needs one factor of (x+5)\left(x+5\right) to be complete. Once we find the LCD, we need to multiply each expression by the form of 11 that will change the denominator to the LCD. What do we mean by " the form of 11"? x+5x+5=1\frac{x+5}{x+5}=1 so multiplying an expression by it will not change it's value. For example, we would need to multiply the expression 6(x+3)(x+4)\large\frac{6}{\left(x+3\right)\left(x+4\right)} by x+5x+5\frac{x+5}{x+5} and the expression 9x(x+4)(x+5)\frac{9x}{\left(x+4\right)\left(x+5\right)} by x+3x+3\frac{x+3}{x+3}. Hopefully this process will become clear after you practice it yourself.  As you look through the examples on this page, try to identify the LCD before you look at the answers. Also, try figuring out which "form of 1" you will need to multiply each expression by so that it has the LCD.

Example

Add the rational expressions: 5x+6y\frac{5}{x}+\frac{6}{y}, and define the domain. State the sum in simplest form.

Answer: First, let's define the domain of each term. Since we have x and y in the denominators, we can say x0, and y0x\ne0 ,\text{ and }y\ne0. Now we have to find the LCD. Since x appears once and y appears once,  the LCD will be xyxy.  We then multiply each expression by the appropriate form of 1 to obtain xyxy as the denominator for each fraction.

5xyy+6yxx5yxy+6xxy\begin{array}{l}\frac{5}{x}\cdot \frac{y}{y}+\frac{6}{y}\cdot \frac{x}{x}\\ \frac{5y}{xy}+\frac{6x}{xy}\end{array}
Now that the expressions have the same denominator, we simply add the numerators to find the sum.
6x+5yxy\frac{6x+5y}{xy}

The domain is x0,y0x\ne0, y\ne0

Answer

5x+6y=(6x+5y)xy[/latex], [latex]x0, and y0\frac{5}{x}+\frac{6}{y}=\frac{(6x+5y)}{xy}[/latex], [latex]x\ne0 ,\text{ and }y\ne0

 

Analysis of the Solution

Multiplying by yy\frac{y}{y} or xx\frac{x}{x} does not change the value of the original expression because any number divided by itself is 11, and multiplying an expression by 11 gives the original expression. Here is one more example of adding rational expressions, but in this case, the expressions have denominators with multi-term polynomials. First, we will factor, then find the LCD. Note that x24x^2-4 is a difference of squares and can be factored using special products.

Example

Simplify2x2x24+xx2\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}, and give the domain. State the result in simplest form.

Answer: Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that x cannot be 22 or 2-2 because the denominators would be 00. (x+2)\left(x+2\right) appears a maximum of one time, as does (x2)\left(x–2\right). This means the LCM is (x+2)(x2)\left(x+2\right)\left(x–2\right).

x24=(x+2)(x2)  x2=x2  x+2=x+2  LCM=(x+2)(x2)\begin{array}{l}x^{2}-4=\left(x+2\right)\left(x-2\right)\\\,\,x-2=x-2\\\,\,x+2=x+2\\\,\,\text{LCM}=\left(x+2\right)\left(x-2\right)\end{array}

The LCM becomes the common denominator. Multiply each expression by the equivalent of 11 that will give it the common denominator.

2x2x24=2x2(x+2)(x2)xx2x+2x+2=x(x+2)(x+2)(x2)\begin{array}{r}\frac{2{{x}^{2}}}{{{x}^{2}}-4}=\frac{2{{x}^{2}}}{(x+2)(x-2)}\\\frac{x}{x-2}\cdot \frac{x+2}{x+2}=\frac{x(x+2)}{(x+2)(x-2)}\end{array}

Rewrite the original problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.

2x2(x+2)(x2)+x(x+2)(x+2)(x2) \displaystyle \frac{2{{x}^{2}}}{(x+2)(x-2)}+\frac{x(x+2)}{(x+2)(x-2)}

Combine the numerators.

2x2+x(x+2)(x+2)(x2)2x2+x2+2x(x+2)(x2) \begin{array}{c}\frac{2{{x}^{2}}+x(x+2)}{(x+2)(x-2)}\\\\\frac{2{{x}^{2}}+{{x}^{2}}+2x}{(x+2)(x-2)}\end{array}

Check for simplest form. Since neither (x+2)\left(x+2\right) nor (x2)\left(x-2\right) is a factor of 3x2+2x3{{x}^{2}}+2x, this expression is in simplest form.

3x2+2x(x+2)(x2)\frac{3{{x}^{2}}+2x}{(x+2)(x-2)}

Answer

2x2x24+xx2=3x2+2x(x+2)(x2)[/latex]  [latex]x2,2 \displaystyle \frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}=\frac{3{{x}^{2}}+2x}{(x+2)(x-2)}[/latex]    [latex] \displaystyle x\ne 2,-2

In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors. https://www.youtube.com/watch?v=CKGpiTE5vIg&feature=youtu.be

Subtracting Rational Expressions

To subtract rational expressions, follow the same process you use to add rational expressions. You will need to be careful with signs, though.

Example

Subtract2t+1t2t2t2\frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}, define the domain. State the difference in simplest form.

Answer: Find the LCD of each expression. t+1t+1 cannot be factored any further, but t2t2{{t}^{2}}-t-2 can be. Note that t cannot be 1-1 or 22 because the denominators would be 00.

t+1=t+1t2t2=(t2)(t+1)\begin{array}{c}t+1=t+1\\t^{2}-t-2=\left(t-2\right)\left(t+1\right)\end{array}

Find the least common multiple. t+1t+1 appears exactly once in both of the expressions, so it will appear once in the least common denominator. t2t–2 also appears once. This means that (t2)(t+1)\left(t-2\right)\left(t+1\right) is the least common multiple. In this case, it is easier to leave the common multiple in terms of the factors, so you will not multiply it out. Use the least common multiple for your new common denominator, it will be the LCD.

t+1=t+1t2t2=(t2)(t+1)LCM:(t+1)(t1)\begin{array}{c}t+1=t+1\\t^{2}-t-2=\left(t-2\right)\left(t+1\right)\\\text{LCM}:\left(t+1\right)\left(t-1\right)\end{array}

Compare each original denominator and the new common denominator. Now rewrite the rational expressions to each have the common denominator of (t+1)(t2)\left(t+1\right)\left(t–2\right). You need to multiply t+1t+1 by t2t–2 to get the LCD, so multiply the entire rational expression by t2t2 \displaystyle \frac{t-2}{t-2}. The second expression already has a denominator of (t+1)(t2)\left(t+1\right)\left(t–2\right), so you do not need to multiply it by anything.

2t+1t2t2=2(t2)(t+1)(t2)   t2t2t2=t2(t+1)(t2) \begin{array}{c}\frac{2}{t+1}\cdot \frac{t-2}{t-2}=\frac{2(t-2)}{(t+1)(t-2)}\\\\\,\,\,\frac{t-2}{{{t}^{2}}-t-2}=\frac{t-2}{(t+1)(t-2)}\end{array}

Then rewrite the subtraction problem with the common denominator.

2(t2)(t+1)(t2)t2(t+1)(t2) \frac{2\left(t-2\right)}{\left(t+1\right)\left(t-2\right)}-\frac{t-2}{\left(t+1\right)\left(t-2\right)}

Subtract the numerators and simplify. Remember that parentheses need to be included around the second (t2)\left(t–2\right) in the numerator because the whole quantity is subtracted. Otherwise you would be subtracting just the tt.

2(t2)(t2)(t+1)(t2)2t4t+2(t+1)(t2)t2(t+1)(t2) \begin{array}{c}\frac{2(t-2)-(t-2)}{(t+1)(t-2)}\\\\\frac{2t-4-t+2}{(t+1)(t-2)}\\\\\frac{t-2}{(t+1)(t-2)}\end{array}

The numerator and denominator have a common factor of t2t–2, so the rational expression can be simplified.

t2(t+1)(t2)=1t+1\large\begin{array}{c}\frac{\cancel{t-2}}{(t+1)\cancel{(t-2)}}\\\\=\frac{1}{t+1}\end{array}

Answer

2t+1t2t2t2=1t+1[/latex], [latex]t1,2 \displaystyle \frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}=\frac{1}{t+1}[/latex],   [latex]t\ne -1,2

In the next example, we will give less instruction.  See if you can find the LCD yourself before you look at the answer.

Example

Subtract the rational expressions: 6x2+4x+42x24\frac{6}{{x}^{2}+4x+4}-\frac{2}{{x}^{2}-4}, and define the domain. State the difference in simplest form.

Answer: Note that the denominator of the first expression is a perfect square trinomial, and the denominator of the second expression is a difference of squares so they can be factored using special products. 6(x+2)22(x+2)(x2)Factor.6(x+2)2x2x22(x+2)(x2)x+2x+2Multiply each fraction to get LCD as denominator.6(x2)(x+2)2(x2)2(x+2)(x+2)2(x2)Multiply.6x12(2x+4)(x+2)2(x2)Apply distributive property.4x16(x+2)2(x2)Subtract.4(x4)(x+2)2(x2)Simplify.\begin{array}{cc}\frac{6}{{\left(x+2\right)}^{2}}-\frac{2}{\left(x+2\right)\left(x - 2\right)}\hfill & \text{Factor}.\hfill \\ \frac{6}{{\left(x+2\right)}^{2}}\cdot \frac{x - 2}{x - 2}-\frac{2}{\left(x+2\right)\left(x - 2\right)}\cdot \frac{x+2}{x+2}\hfill & \text{Multiply each fraction to get LCD as denominator}.\hfill \\ \frac{6\left(x - 2\right)}{{\left(x+2\right)}^{2}\left(x - 2\right)}-\frac{2\left(x+2\right)}{{\left(x+2\right)}^{2}\left(x - 2\right)}\hfill & \text{Multiply}.\hfill \\ \frac{6x - 12-\left(2x+4\right)}{{\left(x+2\right)}^{2}\left(x - 2\right)}\hfill & \text{Apply distributive property}.\hfill \\ \frac{4x - 16}{{\left(x+2\right)}^{2}\left(x - 2\right)}\hfill & \text{Subtract}.\hfill \\ \frac{4\left(x - 4\right)}{{\left(x+2\right)}^{2}\left(x - 2\right)}\hfill & \text{Simplify}.\hfill \end{array}

The domain is x2,2x\ne-2,2

Answer

6x2+4x+42x24=4(x4)(x+2)2(x2)[/latex],  [latex]x2,2\frac{6}{{x}^{2}+4x+4}-\frac{2}{{x}^{2}-4}=\frac{4(x-4)}{(x+2)^2(x-2)}[/latex],   [latex]x\ne-2,2

 

Analysis of the solution

In the last example, the LCD was  (x+2)2(x2)\left(x+2\right)^2\left(x-2\right).  The reason we need to include (x+2)\left(x+2\right) two times is because it appears two times in the expression 6x2+4x+4\frac{6}{{x}^{2}+4x+4}. The video that follows contains an example of subtracting rational expressions. https://www.youtube.com/watch?v=MMlNtCrkakI&feature=youtu.be

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