Lesson 3: Operations on Algebraic Fractions

New Concept

This lesson expands your skills from basic fractions to algebraic ones. You'll learn to multiply, divide, add, and subtract expressions like $\frac{x+1}{x-2}$ by factoring and finding common denominators, just like with numbers.

What’s next

Get ready to master these skills! Soon, you’ll tackle interactive examples and practice problems for multiplying, dividing, and combining algebraic fractions.

Property

If $b \neq 0$ and $d \neq 0$, then

To multiply algebraic fractions:

1. Factor each numerator and denominator.
2. If any factor appears in both a numerator and a denominator, divide out that factor.
3. Multiply the remaining factors of the numerator and the remaining factors of the denominator.
4. Reduce the product if necessary.

Examples

• Multiply $\frac{5x}{2y} \cdot \frac{3y^2}{10x^3}$. We can cancel common factors to get $\frac{1}{2} \cdot \frac{3y}{2x^2} = \frac{3y}{4x^2}$.
• Multiply $8a \cdot \frac{3}{a^2-a}$. First, write $8a$ as $\frac{8a}{1}$, then factor the denominator: $\frac{8a}{1} \cdot \frac{3}{a(a-1)} = \frac{8}{1} \cdot \frac{3}{a-1} = \frac{24}{a-1}$.
• Multiply $\frac{3x+9}{x-5} \cdot \frac{x^2-25}{6x+18}$. Factor everything first: $\frac{3(x+3)}{x-5} \cdot \frac{(x-5)(x+5)}{6(x+3)}$. Cancel common factors to get $\frac{x+5}{2}$.

Explanation

To multiply fractions, you multiply the tops (numerators) together and the bottoms (denominators) together. It is much easier to first cancel any common factors between any numerator and any denominator before you multiply.

Property

If $b, c, d \neq 0$, then

To divide algebraic fractions:

1. Take the reciprocal of the second fraction and change the operation to multiplication.
2. Follow the rules for multiplication of fractions.

Examples

• Divide $\frac{x^2}{4y} \div \frac{x}{8y^2}$. We change to multiplication: $\frac{x^2}{4y} \cdot \frac{8y^2}{x}$. After canceling, we get $x \cdot 2y = 2xy$.
• Divide $\frac{y+3}{y^2-4} \div \frac{2y+6}{y-2}$. Flip the second fraction and multiply: $\frac{y+3}{(y-2)(y+2)} \cdot \frac{y-2}{2(y+3)}$. Canceling leaves $\frac{1}{2(y+2)}$.
• Divide $\frac{z-5}{10z} \div \frac{25-z^2}{5z^2}$. This becomes $\frac{z-5}{10z} \cdot \frac{5z^2}{(5-z)(5+z)}$. Since $z-5 = -1(5-z)$, we can cancel to get $\frac{-1}{2} \cdot \frac{z}{5+z} = \frac{-z}{2(z+5)}$.

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal (flipping it upside down). This trick turns a division problem into a multiplication problem, which you can then solve by factoring and canceling.

Property

If $c \neq 0$, then

To add or subtract like fractions:

1. Add or subtract the numerators.
2. Keep the same denominator.
3. Reduce the sum or difference if necessary.

A subtraction sign in front of a fraction applies to the entire numerator.

Examples

• Add $\frac{3y-2}{y+1} + \frac{y+6}{y+1}$. Combine the numerators: $\frac{(3y-2)+(y+6)}{y+1} = \frac{4y+4}{y+1}$. Factor the numerator to $\frac{4(y+1)}{y+1}$, which reduces to $4$.
• Subtract $\frac{5x-1}{x-4} - \frac{2x+11}{x-4}$. Use parentheses for the second numerator: $\frac{(5x-1)-(2x+11)}{x-4} = \frac{5x-1-2x-11}{x-4} = \frac{3x-12}{x-4}$. This simplifies to $\frac{3(x-4)}{x-4}=3$.
• Add $\frac{a^2}{a^2-9} + \frac{3a}{a^2-9}$. Combine numerators to get $\frac{a^2+3a}{a^2-9} = \frac{a(a+3)}{(a-3)(a+3)}$. This reduces to $\frac{a}{a-3}$.

Explanation

When fractions have the same denominator, they are 'like' pieces. You can simply add or subtract the numerators (the counts) and keep the denominator (the type of piece). Be careful: subtraction applies to the whole numerator that follows.

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator.
To Find the LCD:

1. Factor each denominator completely.
2. Include each different factor in the LCD as many times as it occurs in any one of the given denominators.

Examples

• Find the LCD for $\frac{5}{6x^2y}$ and $\frac{7}{9xy^3}$. The factors are $2, 3^2, x^2, y^3$. The LCD is $2 \cdot 3^2 \cdot x^2 \cdot y^3 = 18x^2y^3$.
• Find the LCD for $\frac{x}{x^2-16}$ and $\frac{3x}{x^2+8x+16}$. The factored denominators are $(x-4)(x+4)$ and $(x+4)^2$. The LCD is $(x-4)(x+4)^2$.
• Find the LCD for $\frac{1}{5a(a-2)^3}$ and $\frac{b}{10a^2(a-2)}$. The factors are $2, 5, a^2, (a-2)^3$. The LCD is $10a^2(a-2)^3$.

Explanation

The LCD is the smallest shared 'target' denominator for unlike fractions. Find it by factoring all denominators and taking the highest power of each unique factor that appears. This ensures all original denominators divide into it evenly.

Property

To add or subtract algebraic fractions:

1. Find the lowest common denominator (LCD) for the fractions.
2. Build each fraction to an equivalent one with the same denominator using the fundamental principle $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$.
3. Add or subtract the resulting like fractions: Add or subtract their numerators, and keep the same denominator.
4. Reduce the sum or difference if necessary.

Examples

• Subtract $\frac{x+1}{8} - \frac{x-2}{12}$. The LCD is $24$. This becomes $\frac{3(x+1)}{24} - \frac{2(x-2)}{24} = \frac{3x+3 - (2x-4)}{24} = \frac{x+7}{24}$.
• Add $\frac{5}{x^2-9} + \frac{2}{x+3}$. The LCD is $(x-3)(x+3)$. The sum is $\frac{5}{(x-3)(x+3)} + \frac{2(x-3)}{(x-3)(x+3)} = \frac{5+2x-6}{(x-3)(x+3)} = \frac{2x-1}{x^2-9}$.
• Add $\frac{m}{m^2-2m} + \frac{3}{m^2-4}$. The denominators are $m(m-2)$ and $(m-2)(m+2)$. The LCD is $m(m-2)(m+2)$. The sum is $\frac{m(m+2)+3m}{m(m-2)(m+2)} = \frac{m^2+2m+3m}{m(m-2)(m+2)} = \frac{m(m+5)}{m(m-2)(m+2)} = \frac{m+5}{(m-2)(m+2)}$.

Explanation

You can't add or subtract fractions with different denominators. First, find the LCD. Then, 'build up' each fraction by multiplying its top and bottom by the missing factors. Now they are like fractions, ready to be combined.