College Algebra - Week 8

Factoring

Factoring is the inverse of multiplication — you reverse an expanded polynomial back to a product of simpler expressions. This week you master every standard factoring technique: GCF extraction, factoring trinomials, special patterns, and factoring completely by combining methods. Factoring is the gateway to solving quadratic equations and simplifying rational expressions.

Learning Objectives

- Factor out the greatest common factor (GCF) from a polynomial. - Factor trinomials of the form x^2 + bx + c. - Factor trinomials of the form ax^2 + bx + c using the AC method or trial and error. - Apply the difference of squares and sum/difference of cubes formulas. - Factor perfect square trinomials. - Factor by grouping when a polynomial has four terms. - Solve quadratic equations by factoring and applying the zero product property.

1. The GCF: Greatest Common Factor

The GCF is the largest expression that divides evenly into every term. Step 1: Find the GCF of the coefficients. Step 2: Find the GCF of the variable parts (lowest power of each variable present). Step 3: Factor out the GCF. Example 1: Factor 12x^3 + 8x^2 - 4x. GCF of 12, 8, 4 is 4. GCF of x^3, x^2, x is x. GCF = 4x. 12x^3 + 8x^2 - 4x = 4x(3x^2 + 2x - 1). Example 2: Factor 6a^3b^2 + 9a^2b^3. GCF = 3a^2b^2. = 3a^2b^2(2a + 3b).

2. Factoring Trinomials: x^2 + bx + c

Find two integers m and n such that m + n = b and m * n = c. Then: x^2 + bx + c = (x + m)(x + n). Example 1: Factor x^2 + 7x + 12. Find m, n with m + n = 7 and m * n = 12. 3 and 4: 3 + 4 = 7 and 3 * 4 = 12. Result: (x + 3)(x + 4). Example 2: Factor x^2 - 5x + 6. Need sum -5, product 6. Both negative: -2 and -3. Result: (x - 2)(x - 3). Example 3: Factor x^2 + 2x - 15. Need sum 2, product -15. Signs differ: 5 and -3. Result: (x + 5)(x - 3).

3. Factoring Trinomials: ax^2 + bx + c

When the leading coefficient a ≠ 1, use the AC method: 1. Multiply a * c. 2. Find factors of (a*c) that sum to b. 3. Split bx into two terms using those factors. 4. Factor by grouping. Example: Factor 6x^2 + 11x + 4. a*c = 24. Find factors of 24 summing to 11: 3 and 8. 6x^2 + 3x + 8x + 4 Group: 3x(2x + 1) + 4(2x + 1) = (3x + 4)(2x + 1) Verify: (3x+4)(2x+1) = 6x^2+3x+8x+4 = 6x^2+11x+4. Correct.

4. Factoring Trinomials: More AC Method Examples

Example: Factor 4x^2 - 4x - 15. a*c = -60. Find factors of -60 summing to -4: -10 and 6. 4x^2 - 10x + 6x - 15 2x(2x - 5) + 3(2x - 5) = (2x + 3)(2x - 5) Example: Factor 2x^2 - 7x + 3. a*c = 6. Find factors summing to -7 with product 6: -1 and -6. 2x^2 - 6x - x + 3 2x(x - 3) - 1(x - 3) = (2x - 1)(x - 3)

5. Special Pattern: Difference of Squares

a^2 - b^2 = (a + b)(a - b) This is one of the fastest patterns to recognize. Examples: x^2 - 9 = (x + 3)(x - 3) 4x^2 - 25 = (2x + 5)(2x - 5) 9a^2 - 16b^2 = (3a + 4b)(3a - 4b) x^4 - 1 = (x^2 + 1)(x^2 - 1) = (x^2 + 1)(x + 1)(x - 1) Note: a SUM of squares (a^2 + b^2) does NOT factor over real numbers.

6. Perfect Square Trinomials

a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a - b)^2 Recognize these by checking: - First and last terms are perfect squares. - Middle term is ±2 * sqrt(first) * sqrt(last). Examples: x^2 + 10x + 25 = (x + 5)^2 x^2 - 6x + 9 = (x - 3)^2 4x^2 + 12x + 9 = (2x + 3)^2 9x^2 - 24x + 16 = (3x - 4)^2

7. Sum and Difference of Cubes

a^3 + b^3 = (a + b)(a^2 - ab + b^2) a^3 - b^3 = (a - b)(a^2 + ab + b^2) Memory device: SOAP — Same, Opposite, Always Positive for the signs. Examples: x^3 + 8 = (x + 2)(x^2 - 2x + 4) x^3 - 27 = (x - 3)(x^2 + 3x + 9) 8x^3 - 125 = (2x - 5)(4x^2 + 10x + 25) The trinomial factor (a^2 ± ab + b^2) does NOT factor further over the reals.

8. Factoring by Grouping

When a polynomial has four terms, try grouping: 1. Split into two groups of two terms. 2. Factor the GCF from each group. 3. Factor out the common binomial. Example: Factor 2x^3 + 3x^2 + 4x + 6. Group: (2x^3 + 3x^2) + (4x + 6) Factor: x^2(2x + 3) + 2(2x + 3) Common factor: (x^2 + 2)(2x + 3) Example 2: Factor ax + bx + ay + by. Group: x(a + b) + y(a + b) = (x + y)(a + b).

9. Factoring Completely

Always factor completely: 1. First, factor out any GCF. 2. Then apply other techniques (trinomial, difference of squares, etc.). 3. Check whether any factor can be factored further. Example: Factor 2x^3 - 18x. GCF = 2x: 2x(x^2 - 9). Factor difference of squares: 2x(x + 3)(x - 3). Example 2: Factor 3x^4 - 12x^2. GCF = 3x^2: 3x^2(x^2 - 4) = 3x^2(x + 2)(x - 2).

10. Solving by Factoring: Zero Product Property

Zero Product Property: If AB = 0, then A = 0 or B = 0. To solve a quadratic equation by factoring: 1. Move all terms to one side (set equation = 0). 2. Factor completely. 3. Set each factor equal to 0. 4. Solve each equation. Example: Solve x^2 + 2x - 15 = 0. Factor: (x + 5)(x - 3) = 0. x + 5 = 0 → x = -5. x - 3 = 0 → x = 3. Solutions: x = -5 or x = 3.

11. Solving Quadratic Equations: More Examples

Example 1: Solve 2x^2 - 8 = 0. 2(x^2 - 4) = 0 → 2(x+2)(x-2) = 0. Solutions: x = 2 or x = -2. Example 2: Solve 3x^2 + 5x = 0. x(3x + 5) = 0. x = 0 or x = -5/3. Example 3: Solve 6x^2 + x - 2 = 0. Factor: (3x + 2)(2x - 1) = 0. x = -2/3 or x = 1/2.

12. Common Mistakes

- Not factoring out the GCF first, making subsequent factoring harder. - Forgetting to check that each factor is fully factored. - Sign errors in trinomial factoring: for x^2 - 5x - 6, the factors are -6 and +1 (not +6 and -1). - x^2 - 9 = (x-3)^2 is WRONG. x^2 - 9 = (x+3)(x-3). - Setting factors equal to something other than zero: (x+3)(x-2) = 6 cannot be split as x+3=6 or x-2=6.