Algebraic Fraction

Algebraic Fraction

An algebraic fraction is a fraction where the numerator, the denominator, or both contain algebraic expressions (terms with variables). They follow the same rules as regular fractions but involve variables like \(x\), \(y\), \(a\), or \(b\). 

Example: If \(ax + b\) and \(cx + d\) are two algebraic expressions, then the fraction is denoted by \(\frac{ax + b}{cx + d}\), where \(cx + d \neq 0\) is known as algebraic fraction.

Key Points on Algebraic Fraction: 

1. The numerator and/or denominator include variables. 

2. The denominator cannot be zero (e.g., in \( \frac{1}{x} \), \(x \neq 0\)). 

3. They can often be simplified by factoring or canceling common terms. 

Examples of Algebraic Fration: 

1. Simple Algebraic Fraction: 

   \( \frac{2x + 5}{3} \) 

   Explanation: The numerator \(2x + 5\) is an algebraic expression, while the denominator is a number. 

2. Fraction with Variable in Denominator: 

   \( \frac{7}{y - 2} \) 

   Explanation: The denominator \(y - 2\) is an algebraic expression. Here, \(y \neq 2\) (since division by zero is undefined). 

3. Simplifiable Algebraic Fraction: 

   \( \frac{x^2 - 9}{x + 3} \) 

   Simplification: Factor the numerator: 

   \( \frac{(x - 3)(x + 3)}{(x + 3)} \). Cancel \(x + 3\) (if \(x \neq -3\)): 

   Simplified form: \(x - 3\). 

Simplification of Algebraic Fractions

For the simplification of algebraic fractions, we first have to identify whether the denominators of the given fractions are the same or different. Then we have to work accordingly. Here are the steps with the example.

1. If the denominators are the same:

Example: Simplify: \(\frac{x}{(x - y)} + \frac{y}{(x - y)}\)

2. If the denominators are different:

Example: Simplify: \(\frac{x}{(x - y)} - \frac{y}{(x + y)}\)

3. If the denominators are different but can be factorized:

Example: Simplify: \(\frac{1}{(x - y)} - \frac{y}{(x^2 – y^2)}\)

Practice Questions Algebraic Fraction

Q1. What is the value of \(\frac{x}{x - 7} - \frac{x}{x - 7} \) ? Write it. [SEE 2080 MP]

Q2. Simplify the following algebraic fractions.

1. \(\frac{1}{x - y} - \frac{y}{xy – y^2}\) [SEE 2080 GP]
2. \(\frac{m^3 + 1}{m^2 – m + 1} + \frac{m^3 - 1}{m^2 + m + 1} \) [SEE 2080 KaP]
3. \(\frac{1}{x^2 – 7x + 12} + \frac{1}{x^2 – 5x + 6}\) [NEB Model 2080 B]