Algebraic Fractions: Examples with Answers

Algebraic fractions are one of the most fundamental concepts in algebra, yet many students find them challenging at first glance. Simply put, an algebraic fraction is a fraction in which the numerator, denominator, or both contain algebraic expressions.

Algebraic fractions follow the same basic rules of arithmetic, but with variables involved. Understanding how to simplify, add, subtract, multiply, and divide algebraic fractions is an essential skill that lays the groundwork for more advanced topics in mathematics.

What Are Algebraic Fractions?

An algebraic fraction is any fraction of the form P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. For example:

• 3x/5
• (x + 2)/(x − 3)
• (2x² + x)/(x + 1)

These expressions follow the same principles as regular fractions, meaning all operations, simplification, addition, subtraction, multiplication, and division, apply in the same way.

Algebraic Fractions Example

To simplify an algebraic fraction, you need to find common factors in the numerator and denominator and cancel them out.

Example: Simplify: 6x²/9x

Solution: Divide both the numerator and the denominator by their common factor, 3x: 6x² ÷ 3x = 2x 9x ÷ 3x = 3

Answer: 2x/3

Example: Simplify: (x² − 4)/(x + 2)

Solution: Factor the numerator using the difference of squares: x² − 4 = (x + 2)(x − 2)

Now cancel the common factor (x + 2): (x + 2)(x − 2)/(x + 2) = x − 2

Answer: x − 2

Multiplying Algebraic Fractions

To multiply algebraic fractions, multiply the numerators together and the denominators together, then simplify.

Example: Multiply: (3x/4) × (8/x²)

Solution: Multiply numerators: 3x × 8 = 24x Multiply denominators: 4 × x² = 4x²

Result: 24x/4x²

Simplify by dividing by 4x: 24x ÷ 4x = 6 4x² ÷ 4x = x

Answer: 6/x

Another Example: Multiply: (x + 3)/(x − 1) × (x² − 1)/(x + 3)

Solution: Factor x² − 1 = (x + 1)(x − 1)

Now cancel common factors: (x + 3) cancels with (x + 3) (x − 1) cancels with (x − 1)

Answer: x + 1

Dividing Algebraic Fractions

To divide algebraic fractions, multiply the first fraction by the reciprocal of the second.

Example: Divide: (4x/3) ÷ (2x²/9)

Solution: Flip the second fraction and multiply: (4x/3) × (9/2x²)

Multiply numerators: 4x × 9 = 36x Multiply denominators: 3 × 2x² = 6x²

Result: 36x/6x²

Simplify: 36x ÷ 6x = 6, 6x² ÷ 6x = x

Answer: 6/x

Adding Algebraic Fractions

To add algebraic fractions, find a common denominator, adjust the numerators, then add and simplify.

Example: Add: 2/x + 3/x

Solution: Since the denominators are the same: (2 + 3)/x = 5/x

Answer: 5/x

Another Example: Add: 1/(x + 1) + 2/(x + 3)

Solution: The common denominator is (x + 1)(x + 3).

Rewrite each fraction: 1/(x + 1) = (x + 3)/[(x + 1)(x + 3)] 2/(x + 3) = 2(x + 1)/[(x + 1)(x + 3)]

Add the numerators: (x + 3) + 2(x + 1) = x + 3 + 2x + 2 = 3x + 5

Answer: (3x + 5)/[(x + 1)(x + 3)]

Subtracting Algebraic Fractions

Subtraction follows the same process as addition, find a common denominator, then subtract the numerators.

Example: Subtract: 5/(x − 2) − 3/(x + 2)

Solution: The common denominator is (x − 2)(x + 2).

Rewrite each fraction: 5/(x − 2) = 5(x + 2)/[(x − 2)(x + 2)] 3/(x + 2) = 3(x − 2)/[(x − 2)(x + 2)]

Subtract the numerators: 5(x + 2) − 3(x − 2) = 5x + 10 − 3x + 6 = 2x + 16

Answer: (2x + 16)/[(x − 2)(x + 2)] which can also be written as 2(x + 8)/[(x − 2)(x + 2)]

Conclusion

Algebraic fractions may seem intimidating at first, but once you understand the underlying rules, they become much more manageable. Whether you are simplifying, multiplying, dividing, adding, or subtracting, the key is always to factor where possible, find common denominators when needed, and cancel only common factors.