7.3: Add and Subtract Rational Expressions (2024)

RATIONAL EXPRESSION ADDITION AND SUBTRACTION

If \(p\), \(q\), and \(r\) are polynomials where \(r\neq 0\), then

\[\dfrac{p}{r}+\dfrac{q}{r}=\dfrac{p+q}{r} \quad \text{and} \quad \dfrac{p}{r}−\dfrac{q}{r}=\dfrac{p−q}{r}\nonumber\]

Example \(\PageIndex{1}\)

Add: \(\dfrac{11x+28}{x+4}+\dfrac{x^2}{x+4}\).

Solution

Since the denominator is \(x+4\), we must exclude the value \(x=−4\).

\(\begin{array} {ll} &\dfrac{11x+28}{x+4}+\dfrac{x^2}{x+4},\space x\neq −4 \\ \begin{array} {l} \text{The fractions have a common denominator,} \\ \text{so add the numerators and place the sum} \\ \text{over the common denominator.} \end{array} &\dfrac{11x+28+x^2}{x+4} \\ & \\ \text{Write the degrees in descending order.} &\dfrac{x^2+11x+28}{x+4} \\ & \\ \text{Factor the numerator.} &\dfrac{(x+4)(x+7)}{x+4} \\ & \\ \text{Simplify by removing common factors.} &\dfrac{\cancel{(x+4)}(x+7)}{\cancel{x+4}} \\ & \\ \text{Simplify.} &x+7 \end{array}\)

The expression simplifies to \(x+7\) but the original expression had a denominator of \(x+4\) so \(x\neq −4\).

Try It \(\PageIndex{2}\)

Simplify: \(\dfrac{9x+14}{x+7}+\dfrac{x^2}{x+7}\).

Answer

\(x+2\)

Try It \(\PageIndex{3}\)

Simplify: \(\dfrac{x^2+8x}{x+5}+\dfrac{15}{x+5}\).

Answer

\(x+3\)

Example \(\PageIndex{4}\)

Subtract: \(\dfrac{5x^2−7x+3}{x^2−3x+18}−\dfrac{4x^2+x−9}{x^2−3x+18}\).

Solution

\(\begin{array} {ll} &\dfrac{5x^2−7x+3}{x^2−3x+18}−\dfrac{4x^2+x−9}{x^2−3x+18} \\ & \\ \begin{array} {l} \text{Subtract the numerators and place the} \\ \text{difference over the common denominator.} \end{array} &\dfrac{5x^2−7x+3−(4x^2+x−9)}{x^2−3x+18} \\ & \\ \text{Distribute the sign in the numerator.} &\dfrac{5x^2−7x+3−4x^2−x+9}{x^2−3x−18} \\ & \\ \text{Combine like terms.} &\dfrac{x^2−8x+12}{x^2−3x−18} \\ & \\ \text{Factor the numerator and the denominator.} &\dfrac{(x−2)(x−6)}{(x+3)(x−6)} \\ & \\ \text{Simplify by removing common factors.} &\dfrac{(x−2)\cancel{(x−6)}}{(x+3)\cancel{(x−6)}} \\ & \\ &(x−2)(x+3) \end{array}\)

Try It \(\PageIndex{5}\)

Subtract: \(\dfrac{4x^2−11x+8}{x^2−3x+2}−\dfrac{3x^2+x−3}{x^2−3x+2}\).

Answer

\(\dfrac{x−11}{x−2}\)

Try It \(\PageIndex{6}\)

Subtract: \(\dfrac{6x^2−x+20}{x^2−81}−\dfrac{5x^2+11x−7}{x^2−81}\).

Answer

\(\dfrac{x−3}{x+9}\)

Example \(\PageIndex{7}\)

Subtract: \(\dfrac{m^2−6m}{m^2−1}−\dfrac{3m+2}{1−m^2}\).

Solution

The denominators are opposites, so multiply the second fraction by \(\dfrac{−1}{−1}\).
Simplify the second fraction.
The denominators are the same. Subtract the numerators.
Distribute.
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

Try It \(\PageIndex{8}\)

Subtract: \(\dfrac{y^2−5y}{y^2−4}−\dfrac{6y−6}{4−y^2}\).

Answer

\(\dfrac{y+3}{y+2}\)

FIND THE LEAST COMMON DENOMINATOR OF RATIONAL EXPRESSIONS.

1. Factor each denominator completely.
2. List the factors of each denominator. Match factors vertically when possible.
3. Bring down the columns by including all factors, but do not include common factors twice.
4. Write the LCD as the product of the factors.

Example \(\PageIndex{10}\)

a. Find the LCD for the expressions \(\dfrac{8}{x^2−2x−3}\), \(\dfrac{3x}{x^2+4x+3}\) and b. rewrite them as equivalent rational expressions with the lowest common denominator.

Solution

a.

Find the LCD for \(\dfrac{8}{x^2−2x−3}\), \(\dfrac{3x}{x^2+4x+3}\).
Factor each denominator completely, lining up common factors. Bring down the columns.
Write the LCD as the product of the factors.

b.

Factor each denominator.
Multiply each denominator by the ‘missing’ LCD factor and multiply each numerator by the same factor.
Simplify the numerators.

Try It \(\PageIndex{11}\)

a. Find the LCD for the expressions \(\dfrac{2}{x^2−x−12}\), \(\dfrac{1}{x^2−16}\) b. rewrite them as equivalent rational expressions with the lowest common denominator.

Answer

a. \((x−4)(x+3)(x+4)\)
b. \(\dfrac{2x+8}{(x−4)(x+3)(x+4)}\),
\(\dfrac{x+3}{(x−4)(x+3)(x+4)}\)

Try It \(\PageIndex{12}\)

a. Find the LCD for the expressions \(\dfrac{3x}{x^2−3x+10}\), \(\dfrac{5}{x^2+3x+2}\) b. rewrite them as equivalent rational expressions with the lowest common denominator.

Answer

a. \((x+2)(x−5)(x+1)\)
b. \(\dfrac{3x^2+3x}{(x+2)(x−5)(x+1)}\),
\(\dfrac{5x−25}{(x+2)(x−5)(x+1)}\)

Example \(\PageIndex{13}\): How to Add Rational Expressions with Unlike Denominators

Add: \(\dfrac{3}{x−3}+\dfrac{2}{x−2}\).

Solution

Try It \(\PageIndex{14}\)

Add: \(\dfrac{2}{x−2}+\dfrac{5}{x+3}\).

Answer

\(\dfrac{7x−4}{(x−2)(x+3)}\)

Try It \(\PageIndex{15}\)

Add: \(\dfrac{4}{m+3}+\dfrac{3}{m+4}\).

Answer

\(\dfrac{7m+25}{(m+3)(m+4)}\)

ADD OR SUBTRACT RATIONAL EXPRESSIONS.

1. Determine if the expressions have a common denominator.
   • Yes – go to step 2.
   • No – Rewrite each rational expression with the LCD.
     • Find the LCD.
     • Rewrite each rational expression as an equivalent rational expression with the LCD.
2. Add or subtract the rational expressions.
3. Simplify, if possible.

Example \(\PageIndex{16}\)

Add: \(\dfrac{8}{x^2−2x−3}+\dfrac{3x}{x^2+4x+3}\).

Solution

Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
\(\begin{array} {ll} & \\ & \\ \text{Find the LCD.} &\begin{array} {l} \hspace{5mm} x^2−2x−3=(x+1)(x−3) \\ \underline{x^2+4x+3=(x+1)\quad (x+3)} \\ & \\ \qquad LCD=(x+1)(x−3)(x+3) \end{array} \end{array} \)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
	The numerator is prime, so there are no common factors.

Try It \(\PageIndex{17}\)

Add: \(\dfrac{1}{m^2−m−2}+\dfrac{5m}{m^2+3m+2}\).

Answer

\(\dfrac{5m^2−9m+2}{(m+1)(m−2)(m+2)}\)

Try It \(\PageIndex{18}\)

Add: \(\dfrac{2n}{n^2−3n−10}+\dfrac{6}{n^2+5n+6}\).

Answer

\(\dfrac{2n^2+12n−30}{(n+2)(n−5)(n+3)}\)

Example \(\PageIndex{19}\)

Subtract: \(\dfrac{8y}{y^2−16}−\dfrac{4}{y−4}\).

Solution

Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
\(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} y^2−16=(y−4)(y+4) \\ \quad \underline{y−4=y−4} \\ LCD=(y−4)(y+4) \end{array} \end{array} \)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Remove common factors
Simplify.

Try It \(\PageIndex{20}\)

Subtract: \(\dfrac{2x}{x^2−4}−\dfrac{1}{x+2}\).

Answer

\(\dfrac{1}{x−2}\)

Try It \(\PageIndex{21}\)

Subtract: \(\dfrac{3}{z+3}−\dfrac{6z}{z^2−9}\).

Answer

\(\dfrac{−3}{z−3}\)

Example \(\PageIndex{22}\)

Subtract: \(\dfrac{−3n−9}{n^2+n−6}−\dfrac{n+3}{2−n}\).

Solution

Factor the denominator.
Since \(n−2\) and \(2−n\) are opposites, we will multiply the second rational expression by \(\dfrac{−1}{−1}\).
Simplify. Remember, \(a−(−b)=a+b\).
Do the rational expressions have a common denominator? No.
\(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} n^2+n−6=(n−2)(n+3) \\ \quad\underline{n−2=(n−2)} \\ LCD=\quad (n−2)(n+3) \end{array} \end{array} \)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Simplify.

Try It \(\PageIndex{23}\)

Subtract: \(\dfrac{3x−1}{x^2−5x−6}−\dfrac{2}{6−x}\).

Answer

\(\dfrac{5x+1}{(x−6)(x+1)}\)

Try It \(\PageIndex{24}\)

Subtract: \(\dfrac{−2y−2}{y^2+2y−8}−\dfrac{y−1}{2−y}\).

Answer

\(\dfrac{y+3}{y+4}\)

Example \(\PageIndex{25}\)

Subtract: \(\dfrac{4}{a^2+6a+5}−\dfrac{3}{a^2+7a+10}\).

Solution

Factor the denominators.
Do the rational expressions have a common denominator? No.
\(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} a^2+6a+5=(a+1)(a+5) \\ \underline{a^2+7a+10=(a+5)(a+2)} \\ LCD=(a+1)(a+5)(a+2) \end{array} \end{array} \)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.
Look for common factors.
Simplify.

Try It \(\PageIndex{26}\)

Subtract: \(\dfrac{3}{b^2−4b−5}−\dfrac{2}{b^2−6b+5}\).

Answer

\(\dfrac{1}{(b+1)(b−1)}\)

Try It \(\PageIndex{27}\)

Subtract: \(\dfrac{4}{x^2−4}−\dfrac{3}{x^2−x−2}\).

Answer

\(\dfrac{1}{(x+2)(x+1)}\)

Example \(\PageIndex{28}\)

Simplify: \(\dfrac{2u}{u−1}+\dfrac{1}{u}−\dfrac{2u−1}{u^2−u}\).

Solution

Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} u−1=(u−1) \\ u=u \\ \underline{u^2−u=u(u−1)} \\ LCD=u(u−1) \end{array} \end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Write as one rational expression.
Simplify.
Factor the numerator, and remove common factors.
Simplify.

Try It \(\PageIndex{29}\)

Simplify: \(\dfrac{v}{v+1}+\dfrac{3}{v−1}−\dfrac{6}{v^2−1}\).

Answer

\(\dfrac{v+3}{v+1}\)

Try It \(\PageIndex{30}\)

Simplify: \(\dfrac{3w}{w+2}+\dfrac{2}{w+7}−\dfrac{17w+4}{w^2+9w+14}\).

Answer

\(\dfrac{3w}{w+7}\)

Example \(\PageIndex{31}\)

Find \(R(x)=f(x)−g(x)\) where \(f(x)=\dfrac{x+5}{x−2}\) and \(g(x)=\dfrac{5x+18}{x^2−4}\).

xolution

Substitute in the functions \(f(x)\), \(g(x)\).
Factor the denominators.
Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} x−2=(x−2) \\ \underline{x^2−4=(x−2)(x+2)} \\ \hspace{4mm} LCD=(x−2)(x+2)\end{array} \end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.
Write as one rational expression.
Simplify.
Factor the numerator, and remove common factors.
Simplify.

Try It \(\PageIndex{32}\)

Find \(R(x)=f(x)−g(x)\) where \(f(x)=\dfrac{x+1}{x+3}\) and \(g(x)=\dfrac{x+17}{x^2−x−12}\).

Answer

\(\dfrac{x−7}{x−4}\)

Try It \(\PageIndex{33}\)

Find \(R(x)=f(x)+g(x)\) where \(f(x)=\dfrac{x−4}{x+3}\) and \(g(x)=\dfrac{4x+6}{x^2−9}\).

Answer

\(\dfrac{x^2−3x+18}{(x+3)(x−3)}\)