Factoring Techniques and Methods

Introduction

• Factoring: The process of breaking down an expression into a product of simpler expressions.
• Greatest Common Factor (GCF): The largest number that divides two or more numbers.

Factoring the Greatest Common Factor

• Example: For the expression 3x + 15, the GCF is 3.
  • Divide each term by 3:
    • 3x ÷ 3 = x
    • 15 ÷ 3 = 5
  • Factored form: 3(x + 5)
• More examples:
  • 7x - 28: GCF is 7, factor as 7(x - 4).
  • 4x^2 + 8x: GCF is 4x, factor as 4x(x + 2).
  • 5x^2 - 15x^3: GCF is 5x^2, factor as 5x^2(1 - 3x).

Factoring by Grouping

• Used for polynomials with four terms.
• Example: x^3 - 4x^2 + 3x - 12.
  • Group terms: (x^3 - 4x^2) + (3x - 12).
  • Factor each group: x^2(x - 4) + 3(x - 4).
  • Combine: (x - 4)(x^2 + 3).

Factoring Trinomials

• When Leading Coefficient is 1:
  • Example: x^2 + 7x + 12.
  • Find numbers that multiply to 12 and add to 7: 3 and 4.
  • Factor as (x + 3)(x + 4).
• When Leading Coefficient is not 1:
  • Example: 2x^2 + 20x + 48.
  • Factor out GCF: 2(x^2 + 10x + 24).
  • Continue factoring trinomial.

Factoring a Trinomial with Non-1 Leading Coefficient

• Multiply leading coefficient by constant term.
• Example: 15x^2 + x - 6.
  • Find two numbers that multiply to -90 and add to 1.
  • Factor by grouping.

Perfect Square Trinomials

• Form: a^2 + 2ab + b^2 = (a + b)^2
• Example: x^2 + 8x + 16 factors to (x + 4)^2.
• Check if perfect square by confirming middle term calculation with the formula.

Difference of Squares

• Form: a^2 - b^2 = (a + b)(a - b)
• Example: x^2 - 25 factors to (x + 5)(x - 5).
• Steps involve identifying perfect squares and applying the formula.

Sums and Differences of Cubes

• Sum: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
• Difference: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• Example: x^3 + 8 factors to (x + 2)(x^2 - 2x + 4).

Solving Equations by Factoring

• Use when the equation is set to zero.
• Apply zero product property.
• Example: 6x^2 - 30x = 0, factor out 6x, solve for x = 0 or x = 5.

Conclusion

• Practice is essential for mastering factoring techniques.
• Different methods apply based on polynomial type and form.
• Factoring is a foundational skill for solving polynomial equations.