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Learn GCF, trinomials, and special patterns
Learn step-by-step with practice exercises built right in.
Factoring is the process of breaking down an expression into a product of simpler expressions (factors).
Think of it as the reverse of multiplying:
Why Factor?
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
We'll cover several methods:
Factor completely: 6x + 12
Step 1: Find the greatest common factor (GCF) of the terms: Factors of 6: 1, 2, 3, 6 Factors of 12: 1, 2, 3, 4, 6, 12 GCF = 6
Step 2: Factor out the GCF: 6x + 12 = 6(x + 2)
Step 3: Check by distributing: 6(x + 2) = 6x + 12 ✓
Answer: 6(x + 2)
Avoid these 3 frequent errors
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Always factor out the GCF first!
Steps:
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
Used for polynomials with four terms.
Steps:
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)
For trinomials where the coefficient of x² is 1.
Find two numbers that:
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:
When the leading coefficient is not 1, use the AC method or trial and error.
AC Method:
Steps:
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)
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)
Patterns:
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
Step 1: Always factor out GCF first
Step 2: Count terms
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)
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
Forgetting to factor out GCF first Always look for GCF before other methods!
Stopping too soon Factor completely - check each factor
Sign errors Be careful with negatives, especially in grouping
Not checking work Always multiply back or substitute to verify
Confusing sum and difference of squares a² + b² does NOT factor (over real numbers) a² - b² = (a + b)(a - b)
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).
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)
| 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) |
Factor: x² + 7x + 12
Step 1: This is in the form x² + bx + c. Find two numbers that:
Step 2: List factor pairs of 12: 1 and 12 → 1 + 12 = 13 ✗ 2 and 6 → 2 + 6 = 8 ✗ 3 and 4 → 3 + 4 = 7 ✓
Step 3: Write the factored form: x² + 7x + 12 = (x + 3)(x + 4)
Step 4: Check using FOIL: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓
Answer: (x + 3)(x + 4)
Factor completely:
Step 1: Find the GCF of the terms
Step 2: Factor out
Check: ✓
Answer:
Factor:
We need two numbers that multiply to 20 and add to -9
List factor pairs of 20:
Answer:
Check: ✓
Factor: x² - 9
Step 1: Recognize this as a difference of squares: a² - b² x² - 9 = x² - 3²
Step 2: Use the formula: a² - b² = (a + b)(a - b) x² - 3² = (x + 3)(x - 3)
Step 3: Check using FOIL: (x + 3)(x - 3) = x² - 3x + 3x - 9 = x² - 9 ✓
Note: The middle terms cancel out, which is characteristic of difference of squares.
Answer: (x + 3)(x - 3)
Factor completely: 2x² + 11x + 12
Step 1: This is in the form ax² + bx + c where a ≠ 1. a = 2, b = 11, c = 12
Step 2: Find two numbers that multiply to ac = 2(12) = 24 and add to b = 11: Factor pairs of 24: 1 and 24 → 1 + 24 = 25 ✗ 2 and 12 → 2 + 12 = 14 ✗ 3 and 8 → 3 + 8 = 11 ✓
Step 3: Rewrite the middle term using 3 and 8: 2x² + 11x + 12 = 2x² + 3x + 8x + 12
Step 4: Factor by grouping: = (2x² + 3x) + (8x + 12) = x(2x + 3) + 4(2x + 3) = (2x + 3)(x + 4)
Step 5: Check using FOIL: (2x + 3)(x + 4) = 2x² + 8x + 3x + 12 = 2x² + 11x + 12 ✓
Answer: (2x + 3)(x + 4)
Factor completely:
Step 1: Factor out the GCF ()
Step 2: Notice is a difference of squares
Step 3: Combine all factors
Answer:
Factor completely: 3x³ - 48x
Step 1: First, look for a GCF: GCF of 3x³ and 48x is 3x
Step 2: Factor out the GCF: 3x³ - 48x = 3x(x² - 16)
Step 3: Check if what remains can be factored further: x² - 16 is a difference of squares: x² - 4²
Step 4: Factor the difference of squares: 3x(x² - 16) = 3x(x + 4)(x - 4)
Step 5: Check by multiplying: 3x(x + 4)(x - 4) = 3x(x² - 16) = 3x³ - 48x ✓
Important: Always factor out the GCF first, then look for other factoring patterns!
Answer: 3x(x + 4)(x - 4)