Adding & Subtracting Algebraic Fractions

Algebraic fractions can seem tricky at first, but they follow the same rules as regular fractions. The key to success is understanding how to find a common denominator and simplify

 

What Are Algebraic Fractions?

An algebraic fraction is a fraction where the numerator (top) or denominator (bottom), or both, contain algebraic expressions. For example: $$\frac{x + 2}{3}, \frac{2}{x - 1}, \frac{x^2 - 1}{x + 3}$$

To add or subtract algebraic fractions, you need a common denominator, just like with numerical fractions.

 

Simplifying Algebraic Fractions

Simplifying an algebraic fraction means reducing it to its simplest form by canceling common factors.

 

Example 1: Simplifying a Linear Fraction

Simplify: $$\frac{x^2 + 2x}{x}$$

Solution:

1. Factorise the numerator: $$x^2 + 2x = x(x + 2)$$
2. Write the fraction as: $$\frac{x(x + 2)}{x}$$
3. Cancel the common factor $x$ (as long as $x \not = 0$): $$\frac{x(x + 2)}{x} = x + 2 \\ Final Answer:  \\ x + 2$$

 

Example 2: Simplifying with Quadratics

Simplify: $$\frac{x^2 -4}{x^2 + 4x + 4}$$

Solution:

1. Factorise the numerator and denominator: $$\text{Numerator: } x^2 - 4 = (x - 2)(x + 2) \\  \text{Denominator: } x^2 + 4x + 4 = (x + 2)(x + 2)$$
2. Write the fraction as: $$\frac{(x -2)(x + 2)}{(x + 2)(x + 2)}$$
3. Cancel one $x + 2$ factor from the numerator and denominator (as long as $x \not = -2$): $$\frac{(x -2)(x + 2)}{(x + 2)(x + 2)} = \frac{x - 2}{x + 2}  \\ \text{Final Answer:  } \\  \frac{x - 2}{x + 2}, x \not = -2$$

 

 

 

Adding Algebraic Fractions

When adding algebraic fractions:

1. Find a common denominator (like with normal fractions).
2. Rewrite each fraction with the common denominator.
3. Add the numerators and simplify.

Example 1:

Adding Fractions with the Same Denominator

Add: $$\frac{x}{3} + \frac{2}{3}$$

Step-by-Step Solution:

1. The denominators are the same $(3)$, so you can add the numerators: $$\frac{x}{3} + \frac{2}{3} = \frac{x + 2}{3} \\ Final Answer: \\ \frac{x + 2}{3}$$

 

Example 2: Adding Fractions with Different Denominators

Add: $$\frac{x}{4} + \frac{3}{6}$$

Step-by-Step Solution:

1. Find the least common denominator (LCD) of 4 and 6, which is 12.
2. Rewrite the fractions with a denominator of 12: $$\frac{x}{4} = \frac{3x}{12},  \frac{3}{6} = \frac{6}{12}$$
3. Add the fractions: $$\frac{3x}{12} + \frac{6}{12} = \frac{3x + 6}{12}$$
4. Simplify if possible (factorise the numerator): $$\frac{3x + 6}{12} = \frac{3(x + 2)}{12} \\ Final Answer: \\ \frac{3(x + 2)}{12}$$

Example 3: Adding Fractions with Different Algebraic Denominators 

Add: $$\frac{1}{x} + \frac{1}{x + 2}$$

Solution:

1. Find the common denominator: $$\text{Common denominator: } x(x + 2)$$
2. Rewrite each fraction with the common denominator: $$\frac{1}{x} = \frac{x + 2}{x(x + 2)},  \frac{1}{x + 2} = \frac{x}{x(x + 2)}$$
3. Add the fractions: $$\frac{x + 2}{x(x + 2)} + \frac{x}{x(x + 2)} = \frac{(x + 2) + x}{x(x + 2)}$$
4. Simplify the numerator: $$\frac{(x + 2) + x}{x(x + 2)} = \frac{2x + 2}{x(x + 2)}$$
5. Factorize the numerator: $$\frac{2x + 2}{x(x + 2)} = \frac{2(x + 1)}{x(x + 2)} \\  \text{Final Answer: } \\  \frac{2(x + 1)}{x(x + 2)}$$

 

 

 

 

Subtracting Algebraic Fractions

Subtracting follows the same steps as adding, but you subtract the numerators instead.

Example 1: Subtracting

Subtract: $$\frac{2x}{5} - \frac{3}{10}$$

Step-by-Step Solution:

1. Find the least common denominator (LCD) of 5 and 10, which is 10.
2. Rewrite the fractions with a denominator of 10: $$\frac{2x}{5} = \frac{4x}{10},  \frac{3}{10} = \frac{3}{10}$$
3. Subtract the fractions: $$\frac{4x}{10} - \frac{3}{10} = \frac{4x - 3}{10} \\ Final Answer: \\  \frac{4x - 3}{10}$$

 

Example 2: Subtracting with Factorisation

Subtract: $$\frac{8}{x + 2} -  \frac{5}{x + 4}$$

Step-by-Step Solution:

1. The denominators are already the same $(x + 2)$, so subtract the numerators: $$\frac{x}{x + 2} - \frac{2}{x + 2} = \frac{x - 2}{x + 2} \\ Final Answer: \\ \frac{x - 2}{x + 2}$$

 

Example 3: Subtracting Fractions with Quadratics

Subtract: $$\frac{x}{x - 3} - \frac{2}{x^2 - 9}$$

Solution:

1. Factorise the quadratic denominator: $$x^2 -9 = (x -3)(x + 3)$$
2. Find the common denominator: $$\text{Common denominator: } (x - 3)(x + 3)$$
3. Rewrite each fraction: $$\frac{x}{x - 3} = \frac{x(x + 3)}{(x - 3)(x + 3)} ,  \frac{2}{x^2 - 9} = \frac{2}{(x - 3)(x + 3)}$$
4. Subtract the fractions: $$\frac{x(x + 3)}{(x -3)(x + 3)} - \frac{2}{(x - 3)(x + 3)} = \frac{x(x + 3) - 2}{(x - 3)(x + 3)}$$
5. Expand the numerator: $$x(x + 3) - 2 = x^2 + 3x - 2$$
6. Write the final fraction: $$\frac{x^2 + 3x - 2}{(x - 3)(x + 3)} \\ \text{Final Answer: } \\  \frac{x^2 + 3x - 2}{(x - 3)(x + 3)}$$