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H33 • Lesson 55 of 105

Factoring Techniques

GCF, trinomials, special products, factoring

High School Essentials • 9-12

Prerequisites: H32

Key Concepts

  • factoring
  • GCF
  • trinomials

Factoring Techniques

Factoring is the reverse of multiplication -- it breaks an expression into a product of simpler factors. This skill is essential for solving quadratic equations, simplifying rational expressions, and analyzing functions. Think of factoring as "un-distributing" or finding what was multiplied together to produce the expression you see.

Greatest Common Factor (GCF)

Always start by checking for a GCF. The GCF is the largest factor that divides evenly into every term of the polynomial.

Worked Example 1 -- GCF Factoring

Factor 12x3 - 8x2 + 4x.

  1. Find the GCF of the coefficients: GCF(12, 8, 4) = 4.
  2. Find the lowest power of x present in all terms: x1.
  3. Factor out 4x: 4x(3x2 - 2x + 1).

Verify by distributing: 4x(3x2) - 4x(2x) + 4x(1) = 12x3 - 8x2 + 4x.

Factoring x2 + bx + c (Leading Coefficient 1)

For trinomials where the leading coefficient is 1, find two numbers that:

Then write: (x + first number)(x + second number).

Worked Example 2 -- Factoring x2 + bx + c

Factor x2 + 7x + 12.

  1. We need two numbers that multiply to 12 and add to 7.
  2. Factor pairs of 12: (1, 12), (2, 6), (3, 4). Check sums: 1+12=13, 2+6=8, 3+4=7.
  3. The pair (3, 4) works. So x2 + 7x + 12 = (x + 3)(x + 4).

Factoring ax2 + bx + c (Leading Coefficient not 1)

When a is not 1, use the AC method:

  1. Multiply a and c to get the product AC.
  2. Find two numbers that multiply to AC and add to b.
  3. Rewrite the middle term using those two numbers.
  4. Factor by grouping.

Worked Example 3 -- AC Method

Factor 6x2 + 11x + 4.

  1. AC = 6 * 4 = 24. Find two numbers that multiply to 24 and add to 11.
  2. The pair is (3, 8): 3 * 8 = 24 and 3 + 8 = 11.
  3. Rewrite: 6x2 + 3x + 8x + 4.
  4. Group and factor: 3x(2x + 1) + 4(2x + 1) = (3x + 4)(2x + 1).

Verify: (3x + 4)(2x + 1) = 6x2 + 3x + 8x + 4 = 6x2 + 11x + 4.

Special Patterns

PatternFormulaExample
Difference of Squaresa2 - b2 = (a + b)(a - b)x2 - 25 = (x + 5)(x - 5)
Perfect Square (sum)a2 + 2ab + b2 = (a + b)2x2 + 6x + 9 = (x + 3)2
Perfect Square (diff)a2 - 2ab + b2 = (a - b)2x2 - 10x + 25 = (x - 5)2

For difference of squares, recognize that both terms must be perfect squares, and they must be subtracted. The sum of two squares (a2 + b2) does not factor over the real numbers.

Common Mistake: Forgetting to Factor Completely

Always check if the factors themselves can be factored further. For example, 2x2 - 8 = 2(x2 - 4) is not fully factored. Since x2 - 4 is a difference of squares, the complete factorization is 2(x + 2)(x - 2).

Factoring Strategy Checklist

When facing any polynomial to factor, follow this order: (1) Factor out the GCF first. (2) Count the terms. Two terms -- check for difference of squares. Three terms -- try trinomial factoring or special patterns. Four terms -- try grouping. (3) Check that each factor is fully factored.

Practice Problems

1. Factor 15x2 - 10x.

Show Solution

GCF is 5x. Factor: 5x(3x - 2).

2. Factor x2 - 5x - 14.

Show Solution

Find two numbers that multiply to -14 and add to -5. The pair is (-7, 2): (-7)(2) = -14 and -7 + 2 = -5. Answer: (x - 7)(x + 2).

3. Factor 4x2 - 49.

Show Solution

This is a difference of squares: (2x)2 - 72 = (2x + 7)(2x - 7).

4. Factor 2x2 + 7x + 3.

Show Solution

AC = 2 * 3 = 6. Two numbers that multiply to 6 and add to 7: (1, 6). Rewrite: 2x2 + x + 6x + 3. Group: x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1).

5. Factor completely: 3x3 - 12x.

Show Solution

GCF is 3x: 3x(x2 - 4). Then x2 - 4 is a difference of squares: 3x(x + 2)(x - 2). Final answer: 3x(x + 2)(x - 2).

Summary

Factoring reverses multiplication to express polynomials as products. Always begin with the GCF. For trinomials with leading coefficient 1, find two numbers that multiply to c and add to b. For leading coefficient not 1, use the AC method and factor by grouping. Recognize the difference of squares pattern (a2 - b2) and perfect square trinomials. Always verify by multiplying your factors back together.

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