Addition and Subtraction of Rational Expressions
Before we begin adding and subtracting rational expressions, we need to be able to find the least common denominator.
The least common denominator (abbreviated LCD) is defined as the least common multiple for all the denominators in a group of fractions or rational expressions.
To find the least common denominator of rational expressions:
Example 1: Find the least common multiple of \(4x^2-16\) and \(6x^3-24x^2+24\).
Before we begin adding and subtracting rational expressions, we need to be able to find the least common denominator.
The least common denominator (abbreviated LCD) is defined as the least common multiple for all the denominators in a group of fractions or rational expressions.
To find the least common denominator of rational expressions:
- Factor each denominator completely.
- Write the least common multiple by writing each factor raised to the highest power it occurs in either polynomial.
Example 1: Find the least common multiple of \(4x^2-16\) and \(6x^3-24x^2+24\).
Factors of \(4x^2-16\):
Factors of \(6x^3-24x^2+24x\): |
\(4\left(x^2-4\right)=2^2\left(x-2\right)\left(x+2\right)\)
\(6x\left(x^2-4x+4\right)=2\cdot3\left(x\right)\left(x-2\right)\left(x-2\right)=2\cdot3\left(x\right)\left(x-2\right)^2\) |
Use the highest power of each factor, so the least common multiple is: \(2^2\cdot3x\left(x+2\right)\left(x-2\right)^2=12x\left(x+2\right)\left(x-2\right)^2\).
Example 2: Find the least common multiple of \(9x^2\) and \(3x^2+3x\).
Example 2: Find the least common multiple of \(9x^2\) and \(3x^2+3x\).
Now that the process of finding a least common denominator has been reviewed, we can look at how to add and subtract rational expressions.
To add or subtract fractions, you need to find the least common denominator. Once the denominators are the same, you add or subtract the numerators, then reduce the fraction if necessary.
For example: \(\frac{2}{3}+\frac{7}{5}\)
The least common denominator is \(15\), so we need to write each fraction over \(15\).
\(\large\frac{2\cdot5}{3\cdot5}+\frac{7\cdot3}{5\cdot3}\)
\(\large\frac{10}{15}+\frac{21}{15}\)
\(\large\frac{10+21}{15}\)
\(\large\frac{31}{15}\)
Although that is a simpler example, the process you will follow to add or subtract rational expressions will be the same.
To Add and Subtract Rational Expressions:
- Find the least common denominator (LCD) as previously described.
- Rewrite each term over the LCD.
- Perform the multiplication in the numerator of each rational expression.
- Combine like terms in the numerator (if it is a subtraction problem, don’t forget to distribute the negative sign to EACH term in the second expression).
- Factor the numerator if necessary.
- Reduce the fraction, if necessary, by dividing out common factors between the numerator and the denominator.
Example 3: Add the rational expressions \(\large{\frac{x^2-4}{x^2+5x-14}+\frac{x+3}{x+7}}\).
\(\begin{align}&\frac{x^2-4}{\left(x+7\right)\left(x-2\right)}+\frac{x+3}{x+7}\ & \ &\text{1) Factor the denominator(s) so you can find the LCD.}\\
&\frac{x^2-4}{\left(x+7\right)\left(x-2\right)}+\frac{\left(x+3\right)\left(x-2\right)}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{2) The LCD is}\ \left(x+7\right)\left(x-2\right)\ \text{. Multiply each numerator and denominator by}\\
& \ & \ &\text{the correct factor.}\\
&\frac{x^2-4}{\left(x+7\right)\left(x-2\right)}+\frac{x^2+x-6}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{3) Perform the multiplication in the numerator(s).}\\
&\frac{x^2-4+x^2+x-6}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{4) Rewrite the numerators over one denominator.}\\
&\frac{2x^2+x-10}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{5) Combine like terms.}\\
&\frac{\left(2x+5\right)\left(x-2\right)}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{6) Factor the numerator.}\\
&\frac{2x+5}{x+7}\ & \ &\text{7) Divide out common factors with the denominator.}\end{align}\)
&\frac{x^2-4}{\left(x+7\right)\left(x-2\right)}+\frac{\left(x+3\right)\left(x-2\right)}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{2) The LCD is}\ \left(x+7\right)\left(x-2\right)\ \text{. Multiply each numerator and denominator by}\\
& \ & \ &\text{the correct factor.}\\
&\frac{x^2-4}{\left(x+7\right)\left(x-2\right)}+\frac{x^2+x-6}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{3) Perform the multiplication in the numerator(s).}\\
&\frac{x^2-4+x^2+x-6}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{4) Rewrite the numerators over one denominator.}\\
&\frac{2x^2+x-10}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{5) Combine like terms.}\\
&\frac{\left(2x+5\right)\left(x-2\right)}{\left(x+7\right)\left(x-2\right)}\ & \ &\text{6) Factor the numerator.}\\
&\frac{2x+5}{x+7}\ & \ &\text{7) Divide out common factors with the denominator.}\end{align}\)
Example 4: Add the rational expressions \(\large{\frac{x}{x^2-9}+\frac{x+1}{x^2+6x+9}}\).
Example 5: Subtract the rational expressions \(\large{\frac{x}{x^2-2x-3}-\frac{x-2}{x^2+x-12}}\).
Complex Fractions
A complex fraction is a fraction that contains fractions in its numerator and/or its denominator. There are two methods that can be used to simplify complex fractions:
- Simplify each numerator and denominator and then divide the fractions
- Multiply each numerator and denominator by the LCD of every fraction then simplify.
Example 6: Simplify the complex fraction \(\Large{\frac{\frac{3}{x+5}}{\frac{2}{x-3}+\frac{1}{x+5}}}\). The video will demonstrate both methods described above.