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Factoring

Definition

Factoring in algebra is the process of breaking down a polynomial expression into a product of simpler expressions (factors). It is the reverse operation of expanding or multiplying polynomials.

Prerequisites

To learn factoring, you should be comfortable with:

  1. Variables & Expressions
  2. Multiplication and Division (especially with integers)
  3. Distributive Property (as part of expansion)
  4. Exponents & Radicals (basic exponent rules)

Learning Objectives

After mastering this topic, you should be able to:

  1. Find the Greatest Common Factor (GCF) of terms in a polynomial.
  2. Factor out the GCF from a polynomial.
  3. Factor trinomials of the form x² + bx + c.
  4. Factor trinomials of the form ax² + bx + c (where a ≠ 1).
  5. Recognize and factor special forms: Difference of Squares (a² - b²), Perfect Square Trinomials (a² + 2ab + b², a² - 2ab + b²).
  6. Factor by grouping for polynomials with four terms.
  7. Factor polynomials completely.

Key Techniques

1. Greatest Common Factor (GCF)

  • Find the largest number and highest power of each variable that divides into all terms.
  • Example: GCF of 6x³y² and 9x²y⁴ is 3x²y².
  • Factoring out GCF: 6x³y² + 9x²y⁴ = 3x²y²(2x + 3y²)

2. Factoring Trinomials (x² + bx + c)

  • Find two numbers that multiply to c and add to b.
  • Example: Factor x² + 7x + 10.
    • Find two numbers that multiply to 10 and add to 7 (These are 5 and 2).
    • Factors are (x + 5)(x + 2).

3. Factoring Trinomials (ax² + bx + c)

  • Trial and Error: Find factors of a and c and test combinations.
  • Grouping Method: Find two numbers that multiply to ac and add to b. Split the middle term (bx) using these numbers, then factor by grouping.
  • Example: Factor 2x² + 7x + 3.
    • ac = 2*3 = 6. Find two numbers that multiply to 6 and add to 7 (These are 6 and 1).
    • Rewrite: 2x² + 6x + 1x + 3
    • Group: (2x² + 6x) + (x + 3)
    • Factor GCF from each group: 2x(x + 3) + 1(x + 3)
    • Factor out common binomial: (x + 3)(2x + 1)

4. Difference of Squares (a² - b²)

  • Factors into (a - b)(a + b).
  • Example: Factor x² - 25 = x² - 5² = (x - 5)(x + 5).

5. Perfect Square Trinomials

  • a² + 2ab + b² = (a + b)²
  • a² - 2ab + b² = (a - b)²
  • Example: Factor x² + 6x + 9 = x² + 2(x)(3) + 3² = (x + 3)².

6. Factoring by Grouping (Four Terms)

  • Group the first two terms and the last two terms. Factor out the GCF from each group. If the remaining binomials match, factor out the common binomial.
  • Example: Factor x³ + 2x² + 5x + 10.
    • Group: (x³ + 2x²) + (5x + 10)
    • Factor GCFs: x²(x + 2) + 5(x + 2)
    • Factor out binomial: (x + 2)(x² + 5)

Examples

Example 1: GCF (Middle School)

Problem: Factor 4a² + 8a. Solution: GCF is 4a. Factored form: 4a(a + 2).

Example 2: Simple Trinomial (Middle School/High School)

Problem: Factor y² - 5y - 14. Solution: Find numbers multiplying to -14 and adding to -5 (These are -7 and 2). Factors: (y - 7)(y + 2).

Example 3: Difference of Squares (High School)

Problem: Factor 9m² - 16. Solution: Recognize as (3m)² - 4². Factors: (3m - 4)(3m + 4).

Example 4: Complex Trinomial (High School)

Problem: Factor 3p² - 10p + 8. Solution: Use grouping. ac = 24. Find numbers multiplying to 24 and adding to -10 (These are -6 and -4). Rewrite: 3p² - 6p - 4p + 8 Group: (3p² - 6p) + (-4p + 8) Factor GCFs: 3p(p - 2) - 4(p - 2) Factor out binomial: (p - 2)(3p - 4).

Common Misconceptions

  1. Forgetting the GCF: Always check for a GCF first before applying other methods.
  2. Sign Errors: Especially when factoring trinomials with negative terms.
  3. Stopping Too Soon: Ensure each factor is factored completely (e.g., factoring x⁴ - 16 as (x²-4)(x²+4) is not complete; (x²-4) factors further).
  4. Incorrectly Factoring Sum of Squares: a² + b² is generally prime over real numbers (cannot be factored using simple binomials).

Applications in SAT

Factoring is a critical skill for the SAT Math section, appearing in:

  1. Solving Quadratic Equations: Factoring is a primary method to find the roots/solutions.
  2. Simplifying Rational Expressions: Factoring numerator and denominator to cancel common factors.
  3. Analyzing Polynomial Functions: Finding x-intercepts by factoring.
  4. Algebraic Manipulation: Simplifying complex expressions.

Advanced Connections

Factoring is fundamental for:

  1. Polynomials - Further operations and analysis.
  2. Quadratic Equations - Solving by factoring.
  3. Rational Functions: Simplifying and finding asymptotes/holes.
  4. Calculus: Simplifying expressions before differentiation or integration.

Practice Problems

  1. Basic: Factor out the GCF: 12x³y - 18xy².
  2. Intermediate: Factor the trinomial: x² + 9x + 20.
  3. Advanced: Factor completely: 2a² - 50.
  4. SAT-Level: If x² - y² = 48 and x + y = 12, what is the value of x - y?