Factoring Techniques in Algebra

Overview

This lecture covers multiple factoring techniques in algebra, including GCF, grouping, trinomials, perfect square trinomials, difference of squares, cubes, and solving equations by factoring.

Factoring the Greatest Common Factor (GCF)

• The GCF is the largest term (number and/or variable) that divides each term in an expression.
• To factor out the GCF, divide each term by the GCF and write the result in parentheses.
• Example: 3x + 15 = 3(x + 5); 7x - 28 = 7(x - 4).
• For variables, use the smallest exponent in all terms for the GCF.

Factoring by Grouping

• Used for polynomials with four terms.
• Group terms into two pairs, factor out the GCF from each pair.
• If both groups produce a common binomial factor, factor it out.
• Example: x³ - 4x² + 3x - 12 = (x² + 3)(x - 4).

Factoring Trinomials (Leading Coefficient 1)

• For ax² + bx + c, find two numbers multiplying to c and adding to b.
• Example: x² + 7x + 12 = (x + 3)(x + 4).
• Adjust sign based on middle term for cases with negatives.

Factoring Trinomials (Leading Coefficient ≠ 1)

• Multiply a and c; find two numbers multiplying to ac and summing to b.
• Replace middle term to create four terms and then factor by grouping.

Factoring Perfect Square Trinomials

• Form: a² + 2ab + b² = (a + b)².
• Take square roots of first/last terms; 2ab should equal the middle term.
• Example: x² + 8x + 16 = (x + 4)².

Factoring the Difference of Squares

• Form: a² - b² = (a + b)(a - b).
• Both terms must be perfect squares.
• Example: x² - 25 = (x + 5)(x - 5).

Factoring Sums and Differences of Cubes

• Sum of cubes: a³ + b³ = (a + b)(a² - ab + b²).
• Difference of cubes: a³ - b³ = (a - b)(a² + ab + b²).
• Use cube roots for identification and substitution.

Advanced Factoring (Combinations)

• Sometimes combine perfect square trinomials and difference of squares.
• Example: Factor each part, then apply difference of squares formula to result.

Solving Equations by Factoring

• Set the factored expression equal to zero.
• Apply the zero product property: If ab = 0, then a = 0 or b = 0.
• Solve each simple equation for the variable.

Key Terms & Definitions

• GCF (Greatest Common Factor) — Largest number/variable dividing all terms.
• Binomial — Polynomial with two terms.
• Trinomial — Polynomial with three terms.
• Perfect Square Trinomial — Trinomial that can be written as (a + b)².
• Difference of Squares — Form a² - b², factors to (a + b)(a - b).
• Sum/Difference of Cubes — Cubic forms with specific factoring formulas.
• Zero Product Property — States if ab = 0, then a = 0 or b = 0.

Action Items / Next Steps

• Practice factoring expressions and polynomials using the methods covered.
• Complete assigned homework problems on factoring.
• Review key formulas for quick recall.