Rational Functions and Operations on Rational Expressions

Rational Functions and Multiplying and Dividing Rational Expressions

Introduction to Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is essential for solving more complex algebraic problems.

• Rational Expression: An expression of the form $\frac{P}{Q}$, where P and Q are polynomials and $Q \neq 0$.

• Examples:

  • $\frac{3x^2 + 2x - 4}{4x - 5}$

  • $\frac{4x + 3y}{2x^2 - 3xy + 4y^2}$

  • $\frac{3x^2}{4}$

Domain of a Rational Expression

The domain of a rational expression is the set of all real numbers for which the denominator is not zero. To find the domain, set the denominator equal to zero and solve for the variable; these values are excluded from the domain.

• Example A: Determine the domain of the following rational functions:

  • (a) $f(x) = \frac{x + 1}{x - 6}$ Domain: All real numbers except $x = 6$.

  • (b) $f(x) = \frac{x^2}{x^2 - 4}$ Domain: All real numbers except $x = 2$ and $x = -2$.

  • (c) $f(x) = \frac{x - 3}{x^2 + 2x - 15}$ Domain: All real numbers except $x = 3$ and $x = -5$.

  • (d) $f(x) = \frac{x}{x^2 + 8}$ Domain: All real numbers (since $x^2 + 8$ is never zero for real $x$).

• Example: For which value(s) is $\frac{y - 5}{y^2 + 4}$ undefined?

  • Set denominator $y^2 + 4 = 0 \implies y^2 = -4$ (no real solutions). Answer: None of these (the expression is defined for all real $y$).

Simplifying Rational Expressions

A rational expression is simplified or reduced to lowest terms when the numerator and denominator have no common factors other than 1. Only common factors (not terms) can be canceled.

• Steps to Simplify:

  1. Factor both the numerator and denominator completely.

  2. Divide out any common factors.

• Examples:

  • (a) $\frac{9}{24} = \frac{3 \times 3}{3 \times 8} = \frac{3}{8}$

  • (b) $\frac{xy - y^2}{3y} = \frac{y(x - y)}{3y} = \frac{x - y}{3}$

• Be Careful: Only factors can be canceled, not terms. For example, $\frac{x^2 - 9}{x - 3}$ can be simplified because $x^2 - 9$ factors as $(x - 3)(x + 3)$.

• Example B: Simplify:

  • (a) $\frac{x^2 - 3x - 18}{x - 6}$ Factor numerator: $(x - 6)(x + 3)$ Simplified: $x + 3$ (for $x \neq 6$)

  • (b) $\frac{4x^4 - 4x^3}{x^4 - x^2}$ Factor numerator: $4x^3(x - 1)$ Factor denominator: $x^2(x^2 - 1) = x^2(x - 1)(x + 1)$ Simplified: $\frac{4x^3}{x^2(x + 1)} = \frac{4x}{x + 1}$ (for $x \neq 0, 1, -1$)

  • (c) $\frac{2 - x}{x - 2}$ $2 - x = -(x - 2)$, so $\frac{2 - x}{x - 2} = -1$ (for $x \neq 2$)

Multiplying Rational Expressions

To multiply rational expressions, multiply the numerators together and the denominators together, then simplify.

• General Rule: $\frac{P}{Q} \cdot \frac{R}{S} = \frac{PR}{QS}$, where $Q \neq 0$ and $S \neq 0$.

• Steps:

  1. Factor all numerators and denominators completely.

  2. Divide out common factors.

  3. Multiply numerators and denominators.

  4. Simplify the result, if possible.

• Example C: Multiply:

  • (a) $\frac{1 + 3n}{2n} \cdot \frac{2n - 4}{3n^2 - 2n - 1}$

  • (b) $\frac{4a^2 - 8a}{ab - 2b + 3a - 6} \cdot \frac{8b + 24}{3a + 6}$

  Additional info: Factor all numerators and denominators before multiplying, then cancel common factors.

Dividing Rational Expressions

To divide rational expressions, multiply the first fraction by the reciprocal of the second.

• General Rule: $\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R} = \frac{PS}{QR}$, where $Q \neq 0$, $S \neq 0$, and $R \neq 0$.

• Steps:

  1. Invert the divisor (second fraction).

  2. Multiply as with multiplication of rational expressions.

  3. Simplify the result, if possible.

• Example D: Divide:

  • (a) $\frac{8m^2}{3m^2 - 12} \div \frac{40}{2 - m}$

  • (b) $\frac{a^2 - a - 6}{a^2 - 81} \div \frac{a^2 - 7a - 18}{4a + 36}$

  Additional info: Factor all numerators and denominators, invert the divisor, then multiply and simplify.

Practice Exercises

1. Find the domain of the function: $R(x) = \frac{3 + 2x}{x^3 + x^2 - 2x}$

   • Set denominator to zero: $x^3 + x^2 - 2x = x(x^2 + x - 2) = x(x + 2)(x - 1) = 0$

   • Domain: All real numbers except $x = 0, -2, 1$

2. Divide and simplify: $\frac{5x^2 - 13x - 6}{9x^2 + 17x - 2} \div \frac{8x^2 - 23x - 3}{8x^2 + 17x + 2}$

   • Invert the divisor and multiply, then factor and simplify as above.

3. Perform the indicated operations:

   • $\frac{5a^2 - 20}{3a^2 - 12a} \div \frac{a^3 + 2a^2}{2a^2 - 8a} \cdot \frac{9a^3 + 6a^2}{2a^2 - 4a}$

   • Factor all numerators and denominators, invert and multiply as appropriate, then simplify.

4. Which of the following is simplified correctly?

   • A. $\frac{x}{x - 3} = -\frac{1}{3}$

   • B. $\frac{3x}{3x - 3} = \frac{x}{x - 1}$

   • C. $\frac{4x}{4x + 4} = 1$

   • D. A and B

   • E. A and C

   • F. B and C

   • G. All

   • H. None

   Additional info: Check each simplification by factoring and reducing as shown in previous sections.

Summary Table: Key Operations with Rational Expressions

Operation	Rule	Key Steps
Finding Domain	Denominator $\neq 0$	Solve denominator for zero; exclude those values
Simplifying	Cancel common factors	Factor numerator and denominator, then reduce
Multiplying	$\frac{P}{Q} \cdot \frac{R}{S} = \frac{PR}{QS}$	Factor, cancel, multiply, simplify
Dividing	$\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R}$	Invert divisor, multiply, simplify