Simplifying Algebraic Fractions

Introduction

• Simplifying algebraic fractions involves factorization.
• Key factorization techniques include:
  • Single bracket factorization
  • Double bracket factorization, including quadratics and difference of two squares.

Single Bracket Factorizations

• Common factor example: Factor out common factors from terms.
  • Example: Factor 3 from 3a + 21 (result: 3(a + 7)).
• Variable factorization: Factor out common variables.
  • Example: Factor b from b² - 5b (result: b(b - 5)).
• Complex examples: Factor out the highest common factor from terms with coefficients and variables.
  • Example: Factor 4ab² from 8ab⁴ and 4a³b².

Double Bracket Factorizations

• Quadratics: Factor quadratic expressions into two brackets.
  • Example: x² - 8x + 12 factors to (x - 6)(x - 2).
• Harder quadratics: When the x² term has a coefficient other than 1.
  • Example: 3x² + 19x + 20 factors to (3x + 4)(x + 5).
• Difference of two squares: Recognize square numbers with no x term.
  • Example: x² - 16 factors to (x + 4)(x - 4).

Simplifying Fractions

• Finding common factors: Factor numerators and denominators to find common terms to cancel.
• Example 1: Simplify (12m + 18) / (2m² + 3m) by factoring and cancelling common terms (result: 6/m).
• Example 2: Simplify expressions with quadratics in denominators by factoring.
  • Recognize and cancel common terms.

Complex Examples

• Difference of squares: Recognize expressions like x² - 9 as a difference of squares.
• Multiple fractions: Multiply fractions after fully factorizing both numerator and denominator.
  • Example: Simplify (5a(4a + 1) / 3(a + 2)) * (6(a² - 4) / (4a + 1)(a - 2)) to 10a.

Division and Addition of Fractions

• Order of operations: Divide fractions before adding them.
• Example: Simplify 9 + (2(2x + 5) / (x³(3x + 1))) ÷ ((x⁴(x - 5)) / (x + 5)(x - 5)) by reversing the second fraction and multiplying.
  • Result: 9 + (2x / (3x + 1)).

Presenting in a Given Form

• Combining fractions: Rewrite expressions over a common denominator when required.
  • Example: Combine fractions into one with a specified denominator, simplify, and match the form ax + b / (cx + d).

Conclusion

• Important to practice factorizing different forms.
• Video resources for further learning.
• Try related exam questions for practice.