Rational Expression

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

\frac{P(x)}{Q(x)}= \frac{Polynomial}{Polynomial}

where P(x) and Q(x) are polynomials, and Q(x) ≠ 0 (since division by zero is undefined). The polynomials P(x) and Q(x) can have any degree, meaning the expression can include terms like x², x³, constants, etc.

A rational expression is a mathematical expression that represents the ratio of two polynomials.

Examples of Rational Expressions:

1) \frac{2x + 3}{x - 1}

2) \frac{x^2 - 4}{x + 2}

3) \frac{3x^3 - 2x + 1}{x^2 + x - 6}

4) \frac{4x^2 + 9}{x^2 - 1}

5) \frac{x^3 + 3x^2 + 3x + 1}{x^2 - 2x + 1}

Not an Rational Expression:

1) \frac{1}{x+sin(x)}

This is not a rational expression because the denominator involves sin⁡(x), which is a trigonometric function, and rational expressions must have polynomials (with integer exponents) in both the numerator and the denominator.

2) \frac{\sqrt{x}}{x+e^{x}}

Not rational because √x​ is a root and e^x is an exponential function, neither of which are polynomials.

Read More about Algebraic Expressions.

How to Simplify 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 \frac{x^2 - 4}{x + 2}.

Solution:

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

Cancel the common factor (x + 2).

Rewrite the simplified expression: \frac{(x - 2)(x + 2)}{x + 2} = x - 2

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

Domain of Rational Expression

The domain of a rational expression refers to all the possible values of x that make the expression valid, i.e., the values of x for which the rational expression does not result in division by zero.

Since a rational expression is a fraction, the denominator plays a crucial role. Division by zero is undefined, so the values of x that make the denominator equal to zero must be excluded from the domain.

How to Find the Domain

1. Identify the Denominator: Look at the denominator of the rational expression.
2. Set the Denominator Equal to Zero: Solve for x by setting the denominator equal to zero. These are the values that cause division by zero.
3. Exclude these Values: The values that make the denominator zero are excluded from the domain of the rational expression.

Example 1: \frac{1}{x-5}

Solution:

Step 1: Identify the denominator, which is x − 5.

Step 2: Set the denominator equal to zero to find the restriction:x − 5 = 0 ⇒ x = 5

Step 3: Exclude x = 5 from the domain, because it makes the denominator zero.

So, the domain of the expression \frac{1}{x - 5}​ is all real numbers except x = 5.

Example 2: \frac{x^2-1}{x^2-4}

Solution:

Step 1: Identify the denominator, which is x² − 4.

Step 2: Set the denominator equal to zero: x² − 4 = 0 ⇒ x² = 4 ⇒ x = 2 or x = −2

Step 3: Exclude x = 2 and x = −2 from the domain because these values make the denominator zero.

So, the domain of the expression \frac{x^2 - 1}{x^2 - 4} is all real numbers except x = 2 and x = −2x.

Finding Roots of Rational Expressions

The roots of a rational expression are the values of x that make the expression equal to zero. A rational expression is a fraction where both the numerator and the denominator are polynomials. To find the roots of the rational expression, you only need to focus on the numerator.

Steps to Find the Roots of a Rational Expressions:

Given a rational expression of the form: P(x)/Q(x):

where P(x) is the numerator polynomial and Q(x) is the denominator polynomial:

1. Set the numerator equal to zero: To find the roots of the rational expression, you solve:P(x) = 0The values of x that make P(x) = 0 are the potential roots of the rational expression.
2. Check the denominator: After finding the potential roots from step 1, make sure that none of these values make the denominator Q(x) equal to zero. If any root of P(x) = 0 also makes Q(x) = 0, that root is not valid because division by zero is undefined.

Example: Consider the rational expression: \frac{x^2-4}{x^2-9}

Solution:

Set the numerator equal to zero: x² - 4 = 0

Solving for x:
x² = 4
x = ±2

Thus, the roots of the numerator are x = 2 and x = -2

Check the denominator: x² − 9 = 0

Substitue x = 2:
2² − 9 = −5 ≠ 0

Substitute x = −2x:
(−2)² − 9 = −5 ≠ 0

Since x = 2 and x = −2 do not make the denominator zero, they are valid roots of the rational expression.

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 \frac{2}{x} + \frac{3}{x+1}.

Solution:

Given: \frac{2}{x} + \frac{3}{x+1}

LCD: x(x+1)

Rewrite: \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)}

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 \frac{3x}{4} \times \frac{2}{x^2}.

Solution:

Multiply: \frac{3x \cdot 2}{4 \cdot x^2} = \frac{6x}{4x^2}

Simplify: \frac{6}{4x} = \frac{3}{2x}

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

Example: Simplify \frac{5}{x} \div \frac{10}{x^2 + x}.

Solution:

Given: \frac{5}{x} \div \frac{10}{x^2 + x}.

Reciprocal: \frac{5}{x} \times \frac{x^2 + x}{10}

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

Simplify: \frac{5(x + 1)}{10} = \frac{x + 1}{2}

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: \frac{x^2 + 3x + 2}{x^3 - 4x + 1}

• 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:

• \frac{x^3 + 5x^2 + 1}{x^2 - x + 3}
  • 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.
• \frac{x^2 + 2x + 1}{x^2 - 4}
  • 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 \frac{6x^2 - 12x}{3x}.

Problem 2: Add \frac{2}{x} + \frac{3}{x+1}.

Problem 3: Subtract \frac{4x}{x^2 - 1} - \frac{2}{x+1}.

Problem 4: Multiply \frac{3x^2}{4} \times \frac{2}{x^2}.

Problem 5: Divide \frac{5x}{x^2 + x} \div \frac{10}{x}.

Problem 6: Simplify \frac{x^2 + 2x - 8}{x^2 - 4}.

Problem 7: Simplify \frac{\frac{3x}{x+2}}{\frac{4}{x+2}}.

Problem 8: Simplify \frac{x^2 - 9}{x^2 + 6x + 9}.

Problem 9: Add \frac{2x}{x^2 - 1} + \frac{x}{x - 1}.

Problem 10: Multiply \frac{x^2 - 1}{x + 1} \times \frac{x + 1}{x - 1}.