Rational Expressions: Fundamental Concepts in College Algebra

Rational Expressions

Definition and Domain

A rational expression is the quotient of two polynomials. The domain of a rational expression is the set of all real numbers for which the expression is defined. Since division by zero is undefined, any value that makes the denominator zero must be excluded from the domain.

• Key Point: To find the domain, set the denominator equal to zero and solve for the variable. Exclude these values from the domain.

• Example: For $\frac{1}{x-7}$, set $x-7=0$ so $x=7$ is excluded from the domain.

Simplifying Rational Expressions

Definition and Procedure

A rational expression is simplified if its numerator and denominator have no common factors other than 1 or -1. The process involves:

• Factoring the numerator and denominator completely.

• Dividing both by any common factors.

Example: Simplify $\frac{x-1}{x+1}$ (with $x \neq -1$). Since there are no common factors, the expression is already simplified.

Multiplying Rational Expressions

Steps for Multiplication

To multiply rational expressions:

• Factor all numerators and denominators completely.

• Divide numerators and denominators by any common factors.

• Multiply the remaining factors in the numerators and denominators.

Example: Multiply $\frac{x+3}{x^2-4} \cdot \frac{x^2-x-6}{x^2+6x+9}$.

• Factor: $x^2-4 = (x-2)(x+2)$, $x^2-x-6 = (x-3)(x+2)$, $x^2+6x+9 = (x+3)^2$.

• After canceling common factors and multiplying, $\frac{x-3}{(x-2)(x+3)}$ (with $x \neq -3, x \neq -2, x \neq 2$).

Dividing Rational Expressions

Steps for Division

To divide rational expressions:

• Rewrite the division as multiplication by the reciprocal of the divisor.

• Factor all numerators and denominators completely.

• Divide out any common factors.

• Multiply the remaining factors.

Example: $\frac{x^2-2x+1}{x^2+x-2} \div \frac{x^2+x}{3x^2+3}$ becomes $\frac{x^2-2x+1}{x^2+x-2} \cdot \frac{3x^2+3}{x^2+x}$.

• Factor: $x^2-2x+1 = (x-1)^2$, $x^2+x-2 = (x+2)(x-1)$, $3x^2+3 = 3(x^2+1)$, $x^2+x = x(x+1)$.

• After canceling common factors, multiply the remaining factors.

Adding and Subtracting Rational Expressions

Same Denominator

When rational expressions have the same denominator:

• Add or subtract the numerators.

• Place the result over the common denominator.

• Simplify if possible.

• Check for excluded values.

Different Denominators

When denominators are different:

• Find the least common denominator (LCD) by factoring each denominator and combining all unique factors.

• Rewrite each expression with the LCD as the denominator.

• Add or subtract the numerators, placing the result over the LCD.

• Simplify if possible.

Example: To add $\frac{1}{x-2}$ and $\frac{2}{x+3}$, the LCD is $(x-2)(x+3)$. Rewrite each fraction with the LCD and add.

Complex Rational Expressions

Definition and Simplification

A complex rational expression (or complex fraction) has a numerator or denominator (or both) that contains rational expressions. To simplify:

• Find the LCD of all rational expressions in the numerator and denominator.

• Multiply both the numerator and denominator by this LCD to clear the fractions.

• Simplify the resulting expression.

Example: Simplify $\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x} - \frac{4}{y}}$. The LCD is $xy$; multiply numerator and denominator by $xy$ and simplify.

Summary Table: Operations with Rational Expressions

Operation	Key Steps	Example
Simplify	Factor numerator and denominator, cancel common factors	$\frac{x^2-1}{x+1} = \frac{(x-1)(x+1)}{x+1} = x-1$ (for $x \neq -1$)
Multiply	Factor all, cancel, multiply remaining factors	$\frac{x}{x+2} \cdot \frac{x+2}{x-3} = \frac{x}{x-3}$ (for $x \neq -2, x \neq 3$)
Divide	Multiply by reciprocal, factor, cancel, multiply	$\frac{x}{x+1} \div \frac{x-1}{x} = \frac{x}{x+1} \cdot \frac{x}{x-1} = \frac{x^2}{(x+1)(x-1)}$
Add/Subtract (same denominator)	Add/subtract numerators, keep denominator	$\frac{2}{x} + \frac{3}{x} = \frac{5}{x}$
Add/Subtract (different denominators)	Find LCD, rewrite, add/subtract numerators	$\frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy}$
Complex Fraction	Multiply numerator and denominator by LCD	$\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x} - \frac{4}{y}}$