Topics
4. Rational Functions & Expressions
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United States of AmericaFL
Grade 912

4.04 Adding and subtracting rational expressions

Lesson
Practice
Lesson

Concept summary

To add two (or more) rational expressions which have the same denominator, we add the numerators to form the new numerator over the common denominator - the same process as for adding fractions:\frac{A}{C} + \frac{B}{C} = \frac{A + B}{C}

Subtracting rational expressions is much the same, but the numerators are subtracted instead of added:\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}

In order to add or subtract rational expressions which have different denominators, we first want to rewrite the expressions such that they all have a common denominator. To do so we find the least common multiple (LCM) of the denominators (sometimes called the least common denominator), then multiply the numerator and denominator of each rational expression by any missing factors:\frac{A}{B} + \frac{C}{D} = \frac{AD}{BD} + \frac{BC}{BD} = \frac{AD + BC}{BD}

Worked examples

Example 1

Fully simplify the expression\frac{k - 4}{3 k} - \frac{k - 22}{3 k}

Approach

These two rational expressions have the same denominator, so we can subtract them by subtracting their numerators (being careful with the signs).

Solution

\displaystyle \frac{k - 4}{3 k} - \frac{k - 22}{3 k}\displaystyle =\displaystyle \frac{\left(k - 4\right) - \left(k - 22\right)}{3 k}Rewrite as a single rational expression
\displaystyle =\displaystyle \frac{k - 4 - k + 22}{3 k}Distribute the subtraction and remove parentheses
\displaystyle =\displaystyle \frac{18}{3 k}Combine like terms in the numerator
\displaystyle =\displaystyle \frac{6}{k}Divide out a common factor of 3

Reflection

Notice that even though neither of the original rational expressions had any common factors to divide out, the result of subtracting them had a common factor of 3.

When adding or subtracting rational expressions, make sure to check for any common factors after the addition or subtraction.

Example 2

Fully simplify the expression\frac{5m}{2p^5} + \frac{4}{p^2m^2}

Approach

These two rational expressions do not have the same denominator, so we will first want to rewrite them to have a common denominator before we add them.

In this case, the denominator will need to have factors of 2, p^5, p^2, and m^2. The smallest expression which does this is 2p^5m^2.

Solution

\displaystyle \frac{5m}{2p^5} + \frac{4}{p^2m^2}\displaystyle =\displaystyle \frac{5m}{2p^5} \cdot \frac{m^2}{m^2} + \frac{4}{p^2m^2} \cdot \frac{2p^3}{2p^3}Multiply each rational expression to create a common denominator
\displaystyle =\displaystyle \frac{5m^3}{2p^5m^2} + \frac{8p^3}{2p^5m^2}Evaluate each product
\displaystyle =\displaystyle \frac{5m^3 + 8p^3}{2p^5m^2}Rewrite as a single rational expression

Reflection

In the original rational expressions, one denominator had a factor of p^5 while the other had a factor of p^2.

Notice that p^2 is already a factor of p^5, however. So the final expression only needed to have a factor of p^5 and not p^7.

We can compare this to adding fractions such as \dfrac{1}{2} and \dfrac{1}{8}. In this case, the LCM will only be 8 and not 16, since 2 is already a factor of 8:\frac{1}{2} + \frac{1}{8} = \frac{4}{8} + \frac{1}{8} = \frac{5}{8}

Though any common multiple will work, using the least common multiple will reduce the amount of simplification required after performing the addition.

Example 3

Fully simplify the expression\frac{y - 2}{6} + \frac{y + 3}{y + 9}

Approach

These two rational expressions do not have the same denominator, so we will first want to rewrite them to have a common denominator before we add them.

In this case, the denominator will need to have factors of 6 and y + 9, and so the least common denominator will be 6\left(y + 9\right).

Solution

\displaystyle \frac{y - 2}{6} + \frac{y + 3}{y + 9}\displaystyle =\displaystyle \frac{y - 2}{6} \cdot \frac{y + 9}{y + 9} + \frac{y + 3}{y + 9} \cdot \frac{6}{6}Multiply each rational expression to create a common denominator
\displaystyle =\displaystyle \frac{y^2 + 7y - 18}{6\left(y + 9\right)} + \frac{6y + 18}{6\left(y + 9\right)}Distribute multiplication in each numerator
\displaystyle =\displaystyle \frac{y^2 + 7y - 18 + 6y + 18}{6\left(y + 9\right)}Rewrite as a single rational expression
\displaystyle =\displaystyle \frac{y^2 + 13y}{6\left(y + 9\right)}Combine like terms in the numerator
\displaystyle =\displaystyle \frac{y\left(y + 13\right)}{6\left(y + 9\right)}Write numerator in factored form

Example 4

Fully simplify the expression\frac{2x + 5}{x^2 - 2x - 3} - \frac{x}{x^2 - 6x + 9}

Approach

These two rational expressions do not have the same denominator, so we want to rewrite them to have a common denominator before we subtract them. In order to do that, we will first factor each denominator.

Solution

Factoring the denominators, we get\frac{2x + 5}{x^2 - 2x - 3} - \frac{x}{x^2 - 6x + 9} = \frac{2x + 5}{\left(x - 3\right)\left(x + 1\right)} - \frac{x}{\left(x - 3\right)^2}So the least common denominator will be \left(x - 3\right)^2\left(x + 1\right).

We can now use this to subtract the two rational expressions:

\displaystyle \frac{2x + 5}{\left(x - 3\right)\left(x + 1\right)} - \frac{x}{\left(x - 3\right)^2}\displaystyle =\displaystyle \frac{2x + 5}{\left(x - 3\right)\left(x + 1\right)} \cdot \frac{x - 3}{x - 3} - \frac{x}{\left(x - 3\right)^2} \cdot \frac{x + 1}{x + 1}Create a common denominator
\displaystyle =\displaystyle \frac{2x^2 - x - 15}{\left(x - 3\right)^2\left(x + 1\right)} - \frac{x^2 + x}{\left(x - 3\right)^2\left(x + 1\right)}Distribute multiplication in each numerator
\displaystyle =\displaystyle \frac{2x^2 - x - 15 - \left(x^2 + x\right)}{\left(x - 3\right)^2\left(x + 1\right)}Rewrite as a single rational expression
\displaystyle =\displaystyle \frac{2x^2 - x - 15 - x^2 - x}{\left(x - 3\right)^2\left(x + 1\right)}Distribute the subtraction
\displaystyle =\displaystyle \frac{x^2 - 2x - 15}{\left(x - 3\right)^2\left(x + 1\right)}Combine like terms in the numerator
\displaystyle =\displaystyle \frac{\left(x - 5\right)\left(x + 3\right)}{\left(x - 3\right)^2\left(x + 1\right)}Factor the numerator

Outcomes

MA.912.AR.1.8

Rewrite a polynomial expression as a product of polynomials over the real or complex number system.

MA.912.AR.1.9

Apply previous understanding of rational number operations to add, subtract, multiply and divide rational algebraic expressions.

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