Lecture on Rational Expressions

Introduction

• Rational expressions are essentially fractions.
• Three steps to solving rational expressions problems:
  1. Factor
  2. State non-permissive values (NPVs)
  3. Simplify

Step 1: Factor

• Always attempt to factor expressions.
• No factoring for purely multiplicative expressions.
• Indicators for factoring include addition or subtraction within expressions.

Step 2: Non-Permissive Values (NPVs)

• NPVs are values that cause division by zero.
• Check all points of division for NPVs.
• Example: For 2 * C = 0, C cannot be zero.
• Express NPVs as X cannot equal [value].*

Step 3: Simplify

• Simplification involves combining fractions and reducing them.
• Combine fractions by multiplying across top and bottom.
• Simplify by reducing resulting fractions:
  1. Reduce numbers to simplest form.
  2. Cancel common factors in numerator and denominator.
  3. Be aware of monomial and binomial distinctions in simplifying.

Multiplication of Rational Expressions

• Combine into one fraction by multiplying numerators and denominators.
• Factor each part before multiplying.
• NPVs must be established before combining.

Division of Rational Expressions

• Convert division to multiplication by flipping the second fraction.
• Follow same steps as multiplication after flipping.

Examples

• Worked through examples with monomials and binomials.
• Highlighted common student mistakes:
  • Cancelling terms incorrectly.
  • Forgetting to state NPVs.

Addition and Subtraction of Rational Expressions

• More complex than multiplication and division.
• Requires common denominators.
• Use factoring to assist in finding common denominators.
• Combine into one fraction by expanding and combining like terms.
• Factor and simplify resulting expression.

Special Considerations

• Beware of complex factoring requirements which may not always be necessary in exams.
• Always double-check NPVs and ensure all possible values are accounted for.
• When simplifying, be cautious not to over-cancel.

Common Pitfalls

• Not recognizing when fully factored or simplified.
• Over-cancelling terms, particularly in complex fractions.
• Missing NPVs, especially in multi-step problems.

Conclusion

• Practice makes perfect with rational expressions.
• Each problem follows the same basic structure.
• Keep organized notes and be systematic in approach.