Lecture Notes on Rational Expressions and Equations

Rational Expressions and Equations

1. Introduction to Rational Expressions

A rational expression is defined as a fraction where both the numerator and the denominator are polynomials. This can be expressed in the form:

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

where \( P(x) \) and \( Q(x) \) are polynomials, and it is crucial that \( Q(x) \neq 0 \). Rational expressions can model a variety of real-world scenarios, making them essential in algebra.

2. Key Properties of Rational Expressions

2.1 Domain

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

1. Set the denominator \( Q(x) \) equal to zero.
2. Solve for \( x \) to find the values that are excluded from the domain.

2.2 Simplifying Rational Expressions

To simplify a rational expression:

1. Factor the numerator and denominator completely.
2. Cancel any common factors.

For example:

Simplify: \( \frac{x^2 - 4}{x^2 - x - 6} \)

Solution:

Factor:

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

Cancel the common factor \( (x + 2) \):

\( \frac{x - 2}{x - 3} \)

3. Operations with Rational Expressions

3.1 Addition and Subtraction

To add or subtract rational expressions:

1. Find a common denominator.
2. Rewrite each expression with the common denominator.
3. Add or subtract the numerators and simplify.

Add: \( \frac{3}{x + 1} + \frac{2}{x - 1} \)

Solution:

Common denominator: \( (x + 1)(x - 1) \)

Rewrite:

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

Simplify:

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

3.2 Multiplication and Division

To multiply rational expressions:

1. Multiply the numerators together.
2. Multiply the denominators together.
3. Simplify if possible.

Multiply: \( \frac{x}{x + 2} \times \frac{x + 2}{x - 2} \)

Solution:

\( \frac{x \cdot (x + 2)}{(x + 2)(x - 2)} \)

Cancel the common factor \( (x + 2) \):

\( \frac{x}{x - 2} \)

To divide rational expressions, multiply by the reciprocal:

Divide: \( \frac{2}{3} \div \frac{4}{5} \)

Solution:

\( \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \)

4. Rational Equations

4.1 Solving Rational Equations

A rational equation is an equation that contains at least one rational expression. To solve a rational equation:

1. Identify the common denominator for all terms.
2. Multiply each term by the common denominator to eliminate the fractions.
3. Simplify and solve the resulting equation.
4. Check for extraneous solutions by substituting back into the original equation.

Solve: \( \frac{1}{x - 1} + \frac{2}{x + 1} = \frac{3}{x^2 - 1} \)

Solution:

Common denominator: \( (x - 1)(x + 1) \)

Multiply through by \( (x - 1)(x + 1) \):

\( (x + 1) + 2(x - 1) = 3 \)

Simplify:

\( x + 1 + 2x - 2 = 3 \)

\( 3x - 1 = 3 \Rightarrow 3x = 4 \Rightarrow x = \frac{4}{3} \)

Check: Substitute \( x = \frac{4}{3} \) back into the original equation to ensure it does not make any denominator zero.

5. Applications of Rational Expressions

Rational expressions are widely used in various fields, including:

• Physics: to express rates and ratios, such as speed and density.
• Economics: for calculating cost per unit, revenue, and profit margins.
• Engineering: to model fluid dynamics, material strengths, and more.

6. Practice Exercises

6.1 Simplification

1. Simplify: \( \frac{x^2 - 1}{x^2 - 3x + 2} \)

2. Simplify: \( \frac{6x^2 + 12x}{3x^2 + 9x} \)

6.2 Addition and Subtraction

3. Add: \( \frac{5}{x - 2} + \frac{3}{x + 2} \)

4. Subtract: \( \frac{4}{x + 1} - \frac{2}{x - 1} \)

6.3 Multiplication and Division

5. Multiply: \( \frac{x - 1}{x + 3} \times \frac{x + 3}{x - 2} \)

6. Divide: \( \frac{2x}{3} \div \frac{4}{5} \)

6.4 Solving Rational Equations

7. Solve: \( \frac{1}{x + 3} = \frac{2}{x - 1} \)

8. Solve: \( \frac{3}{x - 2} + \frac{5}{x + 2} = 1 \)

Rational expressions and equations are vital components of algebra that allow for the representation of complex relationships between variables. Mastery of this topic enables students to approach various mathematical problems and real-world applications with confidence and precision.