Factoring Polynomials

What is Factoring?

Factoring is the process of breaking down an expression into a product of simpler expressions (factors).

Think of it as the reverse of multiplying:

• Multiplying: 3(x + 2) = 3x + 6
• Factoring: 3x + 6 = 3(x + 2)

Why Factor?

• Solving equations (especially quadratics)
• Simplifying expressions
• Finding zeros of functions
• Real-world applications

Factors and Multiples Review

Factors of a number divide evenly into it. Factors of 12: 1, 2, 3, 4, 6, 12

Greatest Common Factor (GCF) is the largest number that divides all terms. GCF of 12 and 18: 6

Types of Factoring

We'll cover several methods:

1. Greatest Common Factor (GCF)
2. Factoring by Grouping
3. Factoring Trinomials (x² + bx + c)
4. Factoring Trinomials (ax² + bx + c where a ≠ 1)
5. Difference of Squares
6. Perfect Square Trinomials

Method 1: Greatest Common Factor (GCF)

Always factor out the GCF first!

Steps:

1. Find the GCF of all terms
2. Divide each term by the GCF
3. Write as GCF times the remaining expression

Example 1: Factor 6x + 12

GCF of 6 and 12 is 6 6x + 12 = 6(x + 2)

Check: 6(x + 2) = 6x + 12 ✓

Example 2: Factor 15x² - 10x

GCF of 15 and 10 is 5 GCF of x² and x is x Overall GCF: 5x

15x² - 10x = 5x(3x - 2)

Check: 5x(3x - 2) = 15x² - 10x ✓

Example 3: Factor 4x³ + 8x² - 12x

GCF: 4x 4x³ + 8x² - 12x = 4x(x² + 2x - 3)

Note: x² + 2x - 3 can be factored further! = 4x(x + 3)(x - 1)

Example 4: Factor -3x² - 6x

Factor out -3x: -3x² - 6x = -3x(x + 2)

Tip: Factor out negative when first term is negative

Method 2: Factoring by Grouping

Used for polynomials with four terms.

Steps:

1. Group terms in pairs
2. Factor GCF from each pair
3. Factor out common binomial

Example 1: Factor x³ + 3x² + 2x + 6

Group: (x³ + 3x²) + (2x + 6)

Factor each group: x²(x + 3) + 2(x + 3)

Factor out (x + 3): (x + 3)(x² + 2)

Check: (x + 3)(x² + 2) = x³ + 2x + 3x² + 6 = x³ + 3x² + 2x + 6 ✓

Example 2: Factor 2x³ - 4x² + 3x - 6

Group: (2x³ - 4x²) + (3x - 6)

Factor: 2x²(x - 2) + 3(x - 2)

Factor out (x - 2): (x - 2)(2x² + 3)

Example 3: Factor 6x² + 9x + 4x + 6

Group: (6x² + 9x) + (4x + 6)

Factor: 3x(2x + 3) + 2(2x + 3)

Factor out (2x + 3): (2x + 3)(3x + 2)

Method 3: Factoring x² + bx + c

For trinomials where the coefficient of x² is 1.

Find two numbers that:

• Multiply to give c (constant term)
• Add to give b (coefficient of x)

Form: x² + bx + c = (x + m)(x + n) where m × n = c and m + n = b

Example 1: Factor x² + 7x + 12

Need two numbers that multiply to 12 and add to 7 Factors of 12: 1×12, 2×6, 3×4 3 + 4 = 7 ✓

x² + 7x + 12 = (x + 3)(x + 4)

Check: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓

Example 2: Factor x² - 5x + 6

Need two numbers that multiply to 6 and add to -5 Both must be negative: -2 and -3

x² - 5x + 6 = (x - 2)(x - 3)

Example 3: Factor x² + 2x - 15

Need two numbers that multiply to -15 and add to 2 One positive, one negative: 5 and -3

x² + 2x - 15 = (x + 5)(x - 3)

Example 4: Factor x² - x - 20

Multiply to -20, add to -1 Factors: -5 and 4

x² - x - 20 = (x - 5)(x + 4)

Sign Pattern:

• Both positive if c > 0 and b > 0
• Both negative if c > 0 and b < 0
• Different signs if c < 0 (larger magnitude has sign of b)

Method 4: Factoring ax² + bx + c (a ≠ 1)

When the leading coefficient is not 1, use the AC method or trial and error.

AC Method:

Steps:

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

Example 1: Factor 2x² + 7x + 3

a = 2, b = 7, c = 3 ac = 2 × 3 = 6

Find two numbers: multiply to 6, add to 7 Numbers: 1 and 6

Rewrite: 2x² + x + 6x + 3 Group: (2x² + x) + (6x + 3) Factor: x(2x + 1) + 3(2x + 1) Result: (2x + 1)(x + 3)

Example 2: Factor 3x² - 10x + 8

ac = 3 × 8 = 24 Find: multiply to 24, add to -10 Numbers: -4 and -6

Rewrite: 3x² - 4x - 6x + 8 Group: (3x² - 4x) + (-6x + 8) Factor: x(3x - 4) - 2(3x - 4) Result: (3x - 4)(x - 2)

Example 3: Factor 6x² + 11x - 10

ac = 6 × (-10) = -60 Find: multiply to -60, add to 11 Numbers: 15 and -4

Rewrite: 6x² + 15x - 4x - 10 Group: (6x² + 15x) + (-4x - 10) Factor: 3x(2x + 5) - 2(2x + 5) Result: (2x + 5)(3x - 2)

Method 5: Difference of Squares

Pattern: a² - b² = (a + b)(a - b)

This only works for difference (subtraction), not sum!

Example 1: Factor x² - 9

Both are perfect squares: x² and 3² x² - 9 = (x + 3)(x - 3)

Example 2: Factor 4x² - 25

Both perfect squares: (2x)² and 5² 4x² - 25 = (2x + 5)(2x - 5)

Example 3: Factor 9x² - 16

(3x)² - 4² = (3x + 4)(3x - 4)

Example 4: Factor x⁴ - 1

(x²)² - 1² = (x² + 1)(x² - 1)

But x² - 1 is also a difference of squares! = (x² + 1)(x + 1)(x - 1)

Example 5: Factor 50x² - 2

First factor out GCF: 2(25x² - 1) Then difference of squares: 2(5x + 1)(5x - 1)

Method 6: Perfect Square Trinomials

Patterns:

• a² + 2ab + b² = (a + b)²
• a² - 2ab + b² = (a - b)²

Recognition: First and last terms are perfect squares, middle term is twice their product.

Example 1: Factor x² + 6x + 9

x² = (x)² 9 = 3² 6x = 2(x)(3) ✓

x² + 6x + 9 = (x + 3)²

Example 2: Factor x² - 10x + 25

x² = (x)² 25 = 5² 10x = 2(x)(5) ✓

x² - 10x + 25 = (x - 5)²

Example 3: Factor 4x² + 12x + 9

4x² = (2x)² 9 = 3² 12x = 2(2x)(3) ✓

4x² + 12x + 9 = (2x + 3)²

Complete Factoring Strategy

Step 1: Always factor out GCF first

Step 2: Count terms

• 2 terms: Difference of squares?
• 3 terms: Trinomial? Perfect square?
• 4 terms: Factor by grouping

Step 3: Check if factoring is complete Can any factor be factored further?

Step 4: Check by multiplying back

Example: Factor completely: 2x³ - 18x

Step 1: GCF is 2x 2x(x² - 9)

Step 2: x² - 9 is difference of squares 2x(x + 3)(x - 3)

Final answer: 2x(x + 3)(x - 3)

Checking Your Work

Method 1: Multiply back Expand your factored form and verify it matches the original

Method 2: Substitute a value Pick x = 2, evaluate both original and factored forms They should give the same result

Common Mistakes to Avoid

1. Forgetting to factor out GCF first Always look for GCF before other methods!

2. Stopping too soon Factor completely - check each factor

3. Sign errors Be careful with negatives, especially in grouping

4. Not checking work Always multiply back or substitute to verify

5. Confusing sum and difference of squares a² + b² does NOT factor (over real numbers) a² - b² = (a + b)(a - b)

When Polynomials Don't Factor

Some polynomials are prime (cannot be factored over integers).

Example: x² + 5x + 1 No integers multiply to 1 and add to 5

This is prime (cannot be factored with integer coefficients).

Real-World Applications

Factoring helps find dimensions, zeros, break-even points.

Example: Area of rectangle is x² + 5x + 6. Find possible dimensions.

Factor: (x + 2)(x + 3)

Dimensions could be (x + 2) by (x + 3)

Quick Reference

| Expression Type | Method | Pattern | |----------------|--------|---------| | 3x + 6 | GCF | a(b + c) | | x² + 5x + 6 | Trinomial (a=1) | (x + m)(x + n) | | 2x² + 5x + 3 | AC Method | Factor by grouping | | x² - 16 | Difference of squares | (x + 4)(x - 4) | | x² + 6x + 9 | Perfect square | (x + 3)² | | x³ + 2x² + 3x + 6 | Grouping | (x + 2)(x² + 3) |

Practice Tips

• Master GCF factoring first
• Memorize perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
• Practice sign patterns for trinomials
• Always check by multiplying
• Try all methods systematically
• Factor completely - don't stop too soon
• Keep practicing until patterns become automatic