# Expressing Rational Expressions

• A rational expression is one that can be expressed as the fraction of two polynomials.
• Working with rational expressions is similar to working with regular fractions.

# Simplifying Rational Expressions

• Always check if top and bottom of the fraction can be factored. If a same factor is found in both, it can be canceled out.
• Example: `(3x^2 -12x)/3x` simplifies as `(3x(x - 4))/3x` and further simplifying gives `x - 4`, after cancelling out `3x`.

# Adding and Subtracting Rational Expressions

• To add or subtract rational expressions, they must have the same denominator, just like regular fractions.
• If two rational expressions do not have the same denominator, find a common denominator and transform them.
• Whenever you perform the operations, reflect the changes in the least common denominator.

# Multiplying and Dividing Rational Expressions

• When multiplying, simply multiply the numerators together and the denominators together. Aim to simplify afterwards.
• When dividing, remember to “flip” the second expression (reciprocal) and then multiply.

# Solving Equations with Rational Expressions

• Similar to equations with fractions, to clear the fraction, multiply by the common denominator.
• Be careful of values that make the denominator zero. Always cross-check with the original equation since these can be extraneous solutions.

# Graphing Rational Functions

• Sketching involves identifying key features like vertical asymptotes, horizontal asymptotes, and holes if any.
• Vertical asymptotes are found where the denominator equals zero, and horizontal asymptotes depend on the degree of the polynomials.

The Golden Rule: Always simplify a rational expression but never cancel terms.

Remember to check your solutions in the original question to ensure they don’t make any denominator equals zero, as they would be undefined.