Factoring Quadratic and Polynomial Expressions: Foundations

[Video: Introduction to Factoring Quadratics]

Learn essential techniques for factoring quadratic and polynomial expressions, a fundamental skill for solving equations on the SAT.

Understanding Factoring

Factoring is the process of breaking down an expression into simpler parts (factors) that when multiplied together give the original expression.

Why Factor?

Solve quadratic equations
Find roots/zeros of functions
Simplify rational expressions
Identify key features of graphs

Factoring Quadratics (ax² + bx + c)

Method 1: Factoring when a = 1

For expressions in the form x² + bx + c:

Find two numbers that multiply to c and add to b
Write as (x + m)(x + n) where m and n are the numbers found

Example: Factor x² + 5x + 6

Find numbers that multiply to 6 and add to 5 → 2 and 3

Solution: (x + 2)(x + 3)

Method 2: Factoring when a ≠ 1 (AC Method)

Multiply a and c (AC)
Find two numbers that multiply to AC and add to b
Rewrite the middle term using these numbers
Factor by grouping

Example: Factor 2x² + 7x + 3

AC = 2×3 = 6. Find numbers: 6 and 1 (6×1=6, 6+1=7)

Rewrite: 2x² + 6x + x + 3

Group: (2x² + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3)

Solution: (2x + 1)(x + 3)

Special Factoring Patterns

Pattern	Form	Factored Form
Difference of Squares	a² - b²	(a + b)(a - b)
Perfect Square Trinomial	a² + 2ab + b²	(a + b)²
Perfect Square Trinomial	a² - 2ab + b²	(a - b)²
Sum/Difference of Cubes	a³ ± b³	(a ± b)(a² ∓ ab + b²)

Factoring by Grouping (for polynomials)

Useful for polynomials with 4 or more terms:

Group terms with common factors
Factor out the GCF from each group
Look for a common binomial factor

Example: Factor x³ + 2x² + 3x + 6

Group: (x³ + 2x²) + (3x + 6)

Factor groups: x²(x + 2) + 3(x + 2)

Common factor: (x + 2)(x² + 3)

Key Takeaways

Always look for GCF first before other methods
Recognize special patterns to factor quickly
Check your factoring by multiplying the factors
When stuck, try the AC method for quadratics
For SAT, focus especially on quadratic factoring

Practice Question

Which of the following is the fully factored form of the expression 3x² - 12x - 36?

A) 3(x - 6)(x + 2)

B) (3x - 6)(x + 6)

C) 3(x + 6)(x - 2)

D) (3x + 6)(x - 6)

Follow these steps:

First factor out the greatest common factor (GCF)
Then factor the remaining quadratic expression
Check your answer by expanding the factored form

Step-by-Step Solution:

1. Factor out GCF (3): 3(x² - 4x - 12)

2. Factor quadratic: Find two numbers that multiply to -12 and add to -4 → -6 and +2

3. Write factored form: 3(x - 6)(x + 2)

4. Verify by expanding: 3[(x)(x) + (x)(2) + (-6)(x) + (-6)(2)] = 3[x² + 2x - 6x - 12] = 3[x² - 4x - 12] = 3x² - 12x - 36 ✓

The correct answer is A) 3(x - 6)(x + 2).