A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. This means it is an algebraic expression that can be written in the form P(x)/Q(x)​, where P(x) and Q(x) are polynomials, and Q(x) ≠ 0. Just like regular fractions, rational expressions can be simplified, added, subtracted, multiplied, and divided.

In this article, we will discuss Rational Expression in detail including operations and solved examples as well.

What is a Rational Expression?

A rational expression is a mathematical expression that represents the ratio of two polynomials. In other words, it is a fraction where the numerator and the denominator are both polynomials. These expressions are similar to rational numbers, which are ratios of integers, but instead of integers, they involve polynomials.

Rational expressions share many similarities with regular fractions and can undergo similar operations such as addition, subtraction, multiplication, and division.

Definition of Rational Expression

A rational expression is an algebraic expression that can be expressed as a fraction, where both the numerator and the denominator are polynomials.

The form of a rational expression is typically written as P(x)/Q(x)​, where P(x) and Q(x) are polynomials and Q(x) ≠ 0. This means the denominator cannot be zero, as division by zero is undefined.

Examples of Rational Expressions

Some examples of rational expressions are:

• [Tex]\frac{2x + 3}{x – 1}[/Tex]
  • This is a simple rational expression with a linear numerator 2x + 3 and a linear denominator x – 1.
• [Tex]\frac{x^2 – 4}{x + 2}[/Tex]
  • The numerator x² – 4 can be factored into (x – 2)(x + 2). The expression simplifies to x – 2 when x ≠ -2.
• [Tex]\frac{3x^3 – 2x + 1}{x^2 + x – 6}[/Tex]
  • This rational expression has a cubic polynomial in the numerator and a quadratic polynomial in the denominator. The denominator can be factored into (x – 2)(x + 3).
• [Tex]\frac{4x^2 + 9}{x^2 – 1}[/Tex]
  • Here, the numerator is a quadratic polynomial, and the denominator can be factored into (x – 1)(x + 1).
• [Tex]\frac{x^3 + 3x^2 + 3x + 1}{x^2 – 2x + 1}[/Tex]
  • The numerator and denominator are both polynomials, with the denominator factoring into (x – 1)².

Read More about Algebraic Expressions.

Simplifying Rational Expressions

To simplify the rational expression, we can use the following steps:

• Step 1: Factorize the numerator and the denominator into their simplest polynomial components.
• Step 2: Identify and Cancel Common Factors.
• Step 3: Rewrite the Simplified Expression.

Let’s consider an example for better understanding.

Example: Simplify [Tex]\frac{x^2 – 4}{x + 2}[/Tex].

Solution:

Factor the numerator: x² – 4 = (x – 2)(x + 2)

Cancel the common factor (x + 2).

Rewrite the simplified expression: [Tex] \frac{(x – 2)(x + 2)}{x + 2} = x – 2 [/Tex]

Result: x – 2 (with the restriction x ≠ -2)

Operations with Rational Expressions

Similar to any other expression, we can perform all the operations on rational expressions i.e.,

• Addition
• Subtraction
• Multiplication
• Division

Addition and Subtraction of Rational Expressions

To add and subtract rational expression, we can use following steps:

Step 1: Identify the least common denominator (LCD) of the rational expressions.

Step 2: Rewrite each expression with the LCD by multiplying the numerator and denominator with the necessary factors for each.

Step 3: Add or subtract the numerators while keeping the common denominator.

Step 4: Simplify the Result.

Let’s consider an example for better understanding.

Example: Simplify [Tex]\frac{2}{x} + \frac{3}{x+1}[/Tex].

Solution:

Given: [Tex]\frac{2}{x} + \frac{3}{x+1}[/Tex]

LCD: x(x+1)

Rewrite: [Tex]\frac{2(x+1)}{x(x+1)} + \frac{3x}{x(x+1)} = \frac{2x + 2 + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)}[/Tex]

Multiplication and Division of Rational Expressions

To multiply rational expressions, we can use following steps:

Step 1: Factor both the numerator and the denominator of each rational expression completely.

Step 2: Multiply the numerators to form the new numerator.

Step 3: Multiply the denominators to form the new denominator.

Step 4: Factor the resulting numerator and denominator if possible and cancel any common factors.

Example: Simplify [Tex]\frac{3x}{4} \times \frac{2}{x^2}[/Tex].

Solution:

Multiply: [Tex]\frac{3x \cdot 2}{4 \cdot x^2} = \frac{6x}{4x^2}[/Tex]

Simplify: [Tex]\frac{6}{4x} = \frac{3}{2x}[/Tex]

To divide rational expressions, we take the reciprocal of the second rational expression (the divisor). After that follow the steps for multiplication.

Example: Simplify [Tex]\frac{5}{x} \div \frac{10}{x^2 + x}[/Tex].

Solution:

Given: [Tex]\frac{5}{x} \div \frac{10}{x^2 + x}[/Tex].

Reciprocal: [Tex]\frac{5}{x} \times \frac{x^2 + x}{10}[/Tex]

Multiply: [Tex]\frac{5(x^2 + x)}{10x} = \frac{5x(x + 1)}{10x}[/Tex]

Simplify: [Tex]\frac{5(x + 1)}{10} = \frac{x + 1}{2}[/Tex]

Proper and Improper Rational Expressions

Rational expressions can be classified into two types based on the degrees of the polynomials in the numerator and the denominator: proper rational expressions and improper rational expressions.

Proper Rational Expressions

A rational expression is considered proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

Example: [Tex]\frac{x^2 + 3x + 2}{x^3 – 4x + 1}[/Tex]

• Here, the degree of the numerator is 2, and the degree of the denominator is 3. Since 2 < 3, this is a proper rational expression.

Improper Rational Expressions

A rational expression is considered improper if the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator.

Examples:

• [Tex]\frac{x^3 + 5x^2 + 1}{x^2 – x + 3}[/Tex]
  • Here, the degree of the numerator is 3, and the degree of the denominator is 2. Since 3 > 2, this is an improper rational expression.
• [Tex]\frac{x^2 + 2x + 1}{x^2 – 4}[/Tex]
  • Here, the degree of the numerator is 2, and the degree of the denominator is also 2. Since the degrees are equal, this is also an improper rational expression.

Read More,

• Addition of Algebraic Expressions
• Subtraction of Algebraic Expression
• Multiplication of Algebraic Expression
• Division of Algebraic Expression
• Factorization of Algebraic Expression

Practice Problems on Rational Expressions

Problem 1: Simplify [Tex]\frac{6x^2 – 12x}{3x}[/Tex].

Problem 2: Add [Tex]\frac{2}{x} + \frac{3}{x+1}[/Tex].

Problem 3: Subtract [Tex]\frac{4x}{x^2 – 1} – \frac{2}{x+1}[/Tex].

Problem 4: Multiply [Tex]\frac{3x^2}{4} \times \frac{2}{x^2}[/Tex].

Problem 5: Divide [Tex]\frac{5x}{x^2 + x} \div \frac{10}{x}[/Tex].

Problem 6: Simplify [Tex]\frac{x^2 + 2x – 8}{x^2 – 4}[/Tex].

Problem 7: Simplify [Tex]\frac{\frac{3x}{x+2}}{\frac{4}{x+2}}[/Tex].

Problem 8: Simplify [Tex] \frac{x^2 – 9}{x^2 + 6x + 9}[/Tex].

Problem 9: Add [Tex]\frac{2x}{x^2 – 1} + \frac{x}{x – 1}[/Tex].

Problem 10: Multiply [Tex] \frac{x^2 – 1}{x + 1} \times \frac{x + 1}{x – 1}[/Tex].

FAQs on Rational Expressions

Define Rational Expression in Math.

A rational expression is a fraction where both the numerator and the denominator are polynomials. It is of the form P(x)/Q(x)​, where P(x) and Q(x)are polynomials, and Q(x) ≠ 0.

How do you Simplify a Rational Expression?

To simplify a rational expression, factor both the numerator and the denominator completely and then cancel any common factors. The goal is to reduce the expression to its lowest terms.

What are Proper and Improper Rational Expressions?

• Proper Rational Expression: The degree of the numerator is less than the degree of the denominator.
• Improper Rational Expression: The degree of the numerator is greater than or equal to the degree of the denominator.