Factoring Methods Overview

Overview

This lecture covers different methods of factoring algebraic expressions, including factoring out the greatest common factor (GCF), factoring by grouping, trinomials, perfect square trinomials, difference of squares, sums and differences of cubes, and solving equations by factoring.

Factoring out the Greatest Common Factor (GCF)

• The GCF is the largest factor shared by all terms in an expression.
• Factor the GCF out by dividing each term and rewriting the expression as GCF × (remaining terms).
• Example: 3x + 15 = 3(x + 5); 7x – 28 = 7(x – 4).
• For variables, factor out the lowest power present in all terms (e.g., GCF of x² and x³ is x²).

Factoring by Grouping

• Used for four-term polynomials.
• Group terms in pairs, factor out the GCF in each group.
• If both groups share a common binomial, factor it out.
• Example: x³ – 4x² + 3x – 12 = (x²(x – 4) + 3(x – 4)) = (x – 4)(x² + 3).

Factoring Trinomials

• When the leading coefficient is 1: Find two numbers that multiply to the constant and add to the middle coefficient.
• Example: x² + 7x + 12 = (x + 3)(x + 4).
• If the leading coefficient is not 1: Multiply the first and last coefficients, find two numbers that multiply to this product and sum to the middle coefficient, rewrite the middle term and factor by grouping.

Perfect Square Trinomials

• Form: a² + 2ab + b² = (a + b)².
• Check by squaring the first and last terms, multiplying them and doubling to match the middle term.
• Example: x² + 8x + 16 = (x + 4)².

Difference of Squares

• Form: a² – b² = (a + b)(a – b).
• Example: x² – 25 = (x + 5)(x – 5).

Sums and Differences of Cubes

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

Combining Factoring Techniques

• For complex expressions, factor out negatives or GCFs first, then use appropriate formulas (difference of squares, trinomials, etc.).
• Sometimes, use perfect square or difference of squares formulas after initial factoring.

Solving Equations by Factoring

• Set the equation to zero, factor the expression, and use the zero-product property: set each factor to zero and solve for the variable.
• Example: x² – 5x – 36 = (x + 4)(x – 9) ⇒ x = –4 or x = 9.

Key Terms & Definitions

• Greatest Common Factor (GCF) — largest factor shared by all terms in an expression.
• Trinomial — a polynomial with three terms.
• Perfect Square Trinomial — a trinomial that can be written as (a + b)² or (a – b)².
• Difference of Squares — an expression in the form a² – b².
• Sum/Difference of Cubes — expressions a³ + b³ or a³ – b³.
• Zero Product Property — if ab = 0, then a = 0 or b = 0.

Action Items / Next Steps

• Practice factoring using each method covered.
• Work on provided example problems and identify method needed for each.
• Review key formulas for factoring trinomials, perfect squares, cubes, and difference of squares.