Notes for Math 3 on Feb. 20th, 2025

Chapter 1: Introduction

• Rational Expressions: Ratio of polynomials.

• Simplifying Rational Expressions:

  • Factor the numerator and denominator completely.

  • Cancel common factors (must be entire factors).

• Operations: Applies traditional fraction multiplication/division rules.

Chapter 2: The Common Factors

• Example: Simplifying ( (x + 2)(x - 4) / ((x - 4)(x + 3)) ).

  • Identify common factors: cancel ( x - 4 ).

  • Result: simplified to ( x + 2 / (x + 3) ).

Chapter 3: A Little Factoring

• Example: Simplifying ( (x - 7) / (x^2 - 49) ).

  • Factor denominator: ( x^2 - 49 ) = ( (x + 7)(x - 7) ).

  • Cancel ( x - 7 ): result is ( 1/(x + 7) ).

Chapter 4: Numerator And Denominator

• Look for clues in the numerator to aid in factoring the denominator.

• Use common factors to simplify at each step.

Chapter 5: Reduce Any Numerator

• Example multiplication: ( (2x^2 - 6x) / (x + 4) \times (x^2 - 16) / x ).

  • Factor: Pull GCF from numerator.

  • Resulting factors: ( (2x)(x - 3)/(x + 4) ) and ( (x + 4)(x - 4)/x ).

  • Cancel common factors: ( x + 4 ) and reduce ( x ).

Chapter 6: Factored Form

• After reduction, multiply across: ( 2(x - 3)(x - 4)/1 ).

• Leave in factored form, don't distribute.

Chapter 7: Leave Behind X

• Example: Factor ( (x^3 - 2x^2 + 2x - 6)/(x^2 + 2) ):

  • Group terms: pull out GCFs.

  • Result: ( (x - 3)(x^2 + 2)/(x^2 + 2) ).

Chapter 8: The Reciprocal Means

• Division transforms to multiplication by reciprocal.

• Identify and cancel common factors.

• Result: multiply across to simplify further.

Chapter 9: Conclusion

• Critical skill: Recognizing and factoring effectively helps in simplifying rational expressions.