Factoring Polynomials Lecture Notes

Overview

In this video, we cover various techniques to factor polynomials, including:

• Taking out the Greatest Common Factor (GCF)
• Difference of Perfect Squares
• Sum and Difference of Perfect Cubes
• Factoring Trinomials using substitution
• Factoring by grouping
• Completing the square
• Using synthetic division
• Advanced examples

Key Concepts

1. Greatest Common Factor (GCF)

• Always look for the GCF first.
• Example: For the binomial 7x + 21, the GCF is 7.
  • Factored form: 7(x + 3)
• Practice with:
  • 8x² + 12x → GCF is 4, resulting in 4(2x + 3).
  • 36x³ + y² - 60x⁴y³ → GCF is 12x²y, resulting in 12x²y(3 - 5xy).

2. Difference of Perfect Squares

• Formula: a² - b² = (a + b)(a - b).
• Example: x² - 25 → x² - 5² = (x + 5)(x - 5).

3. Sum and Difference of Perfect Cubes

• Sum: a³ + b³ = (a + b)(a² - ab + b²).
• Difference: a³ - b³ = (a - b)(a² + ab + b²).
• Example: x³ + 8 → x³ + 2³ = (x + 2)(x² - 2x + 4).

4. Factoring Trinomials

• When the leading coefficient is 1: Find two numbers that multiply to the last term and add to the middle term.
  • Example: x² + 11x + 30 → x² + 5x + 6 → (x + 5)(x + 6).
• When the leading coefficient is not 1 (e.g., 2x² - 5x - 3): Use trial and error or the AC method.

5. Factoring by Grouping

• Group terms with a common factor.
• Example: For x³ + 2x² + x + 2, group as (x³ + 2x²) + (x + 2) → x²(x + 2) + 1(x + 2) → (x + 2)(x² + 1).

6. Completing the Square

• For x² + bx + c, find
  • Half of b, square it, and adjust the equation to maintain equality.
• Example: For x² + 6x + 7, complete the square to get (x + 3)² - 2.

7. Synthetic Division

• Used for polynomials with potential rational roots.
• Example with x³ - 2x² - 5x + 6: Find the roots using synthetic division to simplify.

8. Advanced Factoring Techniques

• Substitution for polynomials like x⁴ + 7x² + 12: Let a = x² to transform into a trinomial.
• Additional practice problems include factoring with imaginary numbers, such as x² + 4 → (x + 2i)(x - 2i), where i = √(-1).

Conclusion

• This lecture covers essential techniques for factoring polynomials that are fundamental in algebra, trigonometry, precalculus, and calculus courses.
• Practice these methods for mastery.