Chapter 6: Rational Expressions – Simplifying, Domains, and Operations

Rational Expressions and Functions

Definition and Structure

A rational expression is a fraction in which both the numerator and the denominator are polynomials. A rational function is a function defined by a rational expression. The general form is:

• Rational Expression: $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$

• Rational Number: A special case where both $P$ and $Q$ are integers and $Q \neq 0$

Key Point: The denominator of a rational expression or function cannot be zero, as division by zero is undefined.

Domain of Rational Expressions

The domain of a rational expression or function is the set of all real numbers except those that make the denominator zero. To find the domain:

• Set the denominator equal to zero and solve for the variable.

• Exclude these values from the domain.

Example: For $f(x) = \frac{x-1}{2x-6}$, set $2x-6=0$ to find $x=3$ is excluded from the domain.

Simplifying Rational Expressions

Process of Simplification

To simplify a rational expression:

• Factor the numerator and denominator completely.

• Cancel any common factors that appear in both the numerator and denominator.

Example: Simplify $\frac{7 \cdot 4}{7 \cdot 5}$:

• Cancel the common factor of 7:

$\frac{7 \cdot 4}{7 \cdot 5} = \frac{4}{5}$

Least Common Denominator (LCD)

Finding the LCD

The Least Common Denominator (LCD) of two or more rational expressions is the smallest expression that each denominator divides into. To find the LCD:

• Factor each denominator completely.

• Identify all unique prime factors.

• Multiply each factor the greatest number of times it occurs in any denominator.

Example: For denominators $x^2 - 9$ and $x^2 - x - 6$:

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

• LCD: $(x+3)(x-3)(x+2)$

Writing Equivalent Expressions with Common Denominators

To add or subtract rational expressions, first rewrite each with the LCD as the denominator:

• Multiply the numerator and denominator of each expression by any missing factors of the LCD.

Example: Rewrite $\frac{5}{x}$ and $\frac{7}{x-3}$ with LCD $(x)(x-3)$:

• $\frac{5}{x} = \frac{5(x-3)}{x(x-3)}$

• $\frac{7}{x-3} = \frac{7x}{x(x-3)}$

Adding and Subtracting Rational Expressions

With Unlike Denominators

To add or subtract rational expressions with different denominators:

• Find the LCD.

• Rewrite each expression with the LCD as the denominator.

• Add or subtract the numerators, keeping the common denominator.

• Simplify the result if possible.

Example: $\frac{5}{x} + \frac{7}{x-3} = \frac{5(x-3) + 7x}{x(x-3)}$

Rational Equations

Definition and Solution Process

A rational equation is an equation that contains one or more rational expressions. To solve a rational equation:

• Determine restrictions by setting each denominator equal to zero and solving for the variable.

• Multiply both sides by the LCD to clear denominators.

• Solve the resulting linear or quadratic equation.

• Check all solutions against the original restrictions; discard any that make a denominator zero.

Example: Solve $\frac{3}{x} = 6$:

• Restriction: $x \neq 0$

• Multiply both sides by $x$: $3 = 6x$

• Solve: $x = \frac{1}{2}$

• Check: $x = \frac{1}{2}$ does not violate the restriction.

Special Cases

If a solution equals a restriction (i.e., makes any denominator zero), it is not a valid solution and should be excluded from the solution set.

Summary Table: Key Concepts in Rational Expressions

Concept	Definition/Process	Example
Rational Expression	Fraction with polynomials in numerator and denominator	$\frac{x+1}{x^2-9}$
Domain	All real numbers except those making denominator zero	$x^2-9=0 \Rightarrow x=3, -3$ excluded
Simplification	Factor and cancel common factors	$\frac{7 \cdot 4}{7 \cdot 5} = \frac{4}{5}$
LCD	Product of all unique prime factors in denominators	$(x+3)(x-3)(x+2)$
Adding/Subtracting	Rewrite with LCD, combine numerators	$\frac{5}{x} + \frac{7}{x-3} = \frac{5(x-3) + 7x}{x(x-3)}$
Rational Equation	Equation with rational expressions	$\frac{3}{x} = 6$