Overview
This lecture covers several techniques for factoring polynomials, including recognizing special forms (such as perfect squares and cubes), factoring by grouping, and dealing with exponents. These skills are fundamental for simplifying expressions and solving polynomial equations.
Factoring the Greatest Common Factor (GCF)
- Always check for the GCF (largest common factor of all terms) before other factoring methods.
- Factor out the GCF by dividing each term by it and expressing the polynomial as GCF × (remaining terms).
- GCF applies to both coefficients and variables.
Factoring Trinomials
- For trinomials of the form x² + bx + c, find two numbers that multiply to c and add to b.
- Factor as (x + p)(x + q) where p × q = c and p + q = b.
- Not all trinomials are factorable; some are prime.
Factoring by Grouping
- Used for trinomials with leading coefficients other than 1.
- Find factors of a × c that sum to b, split the middle term, group, and factor out GCFs from each group.
- Factor pairs and then factor out the common binomial.
Factoring Perfect Square Trinomials
- Perfect square trinomials fit the pattern a² + 2ab + b² = (a + b)².
- First and last terms are perfect squares, and the middle term is twice the product of their roots.
Factoring a Difference of Squares
- A difference of squares appears as a² – b² and factors into (a + b)(a – b).
- Both first and last terms must be perfect squares.
- A sum of squares cannot be factored using real numbers.
Factoring Sums and Differences of Cubes
- The sum of cubes: a³ + b³ = (a + b)(a² – ab + b²).
- The difference of cubes: a³ – b³ = (a – b)(a² + ab + b²).
- Use the SOAP acronym: Same, Opposite, Always Positive for the trinomial’s signs.
Factoring with Fractional or Negative Exponents
- Look for the lowest exponent common to all terms and factor it out.
- Apply the same rules as with integer exponents.
Key Terms & Definitions
- Greatest Common Factor (GCF) — the largest expression that divides all terms of a polynomial.
- Trinomial — a polynomial with three terms.
- Perfect Square Trinomial — a trinomial that is the square of a binomial.
- Difference of Squares — an expression of the form a² – b².
- Sum/Difference of Cubes — expressions of the form a³ + b³ or a³ – b³.
Action Items / Next Steps
- Complete section exercises: factor given polynomials using appropriate techniques.
- Practice identifying and applying specific factoring methods for each type of polynomial.
- Review special forms (perfect squares, cubes) and apply to real-world problems as shown in the exercises.