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Using the distributive property and FOIL method
Learn step-by-step with practice exercises built right in.
Unlike addition and subtraction where we combine like terms, multiplication creates new terms by multiplying each term in one polynomial by each term in the other.
Key Principle: The distributive property a(b + c) = ab + ac
This extends to polynomials of any size!
A monomial is a single term.
Rules:
Example 1: 3x · 5x = (3 · 5)(x · x) = 15x²
Example 2: (2x²)(4x³) = (2 · 4)(x² · x³) = 8x⁵
Example 3: (3x²y)(5xy³) = (3 · 5)(x² · x)(y · y³) = 15x³y⁴
Example 4: (-4a²b)(3ab³) = (-4 · 3)(a² · a)(b · b³) = -12a³b⁴
Use the distributive property: multiply the monomial by each term.
Example 1: 3(x + 5) = 3x + 15
Example 2: 2x(3x + 4) = 2x · 3x + 2x · 4 = 6x² + 8x
Example 3: 5x²(2x² - 3x + 4) = 5x² · 2x² + 5x² · (-3x) + 5x² · 4 = 10x⁴ - 15x³ + 20x²
Example 4: -3x(4x² - 2x + 5) = -3x · 4x² + (-3x) · (-2x) + (-3x) · 5 = -12x³ + 6x² - 15x
Example 5: 2xy(3x²y - 4xy + y²) = 6x³y² - 8x²y² + 2xy³
FOIL stands for: First, Outer, Inner, Last
For (a + b)(c + d):
Then combine like terms.
Example 1: (x + 3)(x + 5)
F: x · x = x² O: x · 5 = 5x I: 3 · x = 3x L: 3 · 5 = 15
Combine: x² + 5x + 3x + 15 = x² + 8x + 15
Example 2: (x + 4)(x - 2)
F: x · x = x² O: x · (-2) = -2x I: 4 · x = 4x L: 4 · (-2) = -8
Combine: x² - 2x + 4x - 8 = x² + 2x - 8
Example 3: (2x + 3)(x + 5)
F: 2x · x = 2x² O: 2x · 5 = 10x I: 3 · x = 3x L: 3 · 5 = 15
Combine: 2x² + 10x + 3x + 15 = 2x² + 13x + 15
Example 4: (3x - 2)(2x - 5)
F: 3x · 2x = 6x² O: 3x · (-5) = -15x I: (-2) · 2x = -4x L: (-2) · (-5) = 10
Combine: 6x² - 15x - 4x + 10 = 6x² - 19x + 10
Draw a 2×2 grid and multiply.
Example: (x + 3)(x + 5)
Create a box with:
Sum all boxes: x² + 5x + 3x + 15 = x² + 8x + 15
(a + b)(a - b) = a² - b²
The outer and inner terms cancel!
Example 1: (x + 5)(x - 5) = x² - 25
Example 2: (3x + 4)(3x - 4) = (3x)² - 4² = 9x² - 16
Example 3: (2x + 7)(2x - 7) = 4x² - 49
Why it works: (x + 5)(x - 5) = x² - 5x + 5x - 25 = x² - 25 (middle terms cancel)
(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b²
Example 1: (x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9
Example 2: (x - 5)² = x² - 2(x)(5) + 5² = x² - 10x + 25
Example 3: (2x + 4)² = (2x)² + 2(2x)(4) + 4² = 4x² + 16x + 16
Example 4: (3x - 2)² = (3x)² - 2(3x)(2) + 2² = 9x² - 12x + 4
Common Mistake: (x + 3)² ≠ x² + 9 You MUST include the middle term: x² + 6x + 9
Distribute each term of the binomial to all terms of the trinomial.
Example 1: (x + 2)(x² + 3x + 4)
x(x² + 3x + 4) + 2(x² + 3x + 4) = x³ + 3x² + 4x + 2x² + 6x + 8 = x³ + 5x² + 10x + 8
Example 2: (2x - 1)(x² - x + 3)
2x(x² - x + 3) - 1(x² - x + 3) = 2x³ - 2x² + 6x - x² + x - 3 = 2x³ - 3x² + 7x - 3
Example 3: (x + 3)(2x² - 5x + 1)
= x(2x² - 5x + 1) + 3(2x² - 5x + 1) = 2x³ - 5x² + x + 6x² - 15x + 3 = 2x³ + x² - 14x + 3
Useful for trinomial × trinomial or larger.
Example: (x + 2)(x² + 3x + 4)
Create a box with x² 3x and 4 across the top, x and 2 down the side:
Sum: x³ + 3x² + 4x + 2x² + 6x + 8 = x³ + 5x² + 10x + 8
Distribute systematically - each term times each term.
Example: (x + 1)(x + 2)(x + 3)
First multiply (x + 1)(x + 2): = x² + 2x + x + 2 = x² + 3x + 2
Then multiply result by (x + 3): (x² + 3x + 2)(x + 3) = x(x² + 3x + 2) + 3(x² + 3x + 2) = x³ + 3x² + 2x + 3x² + 9x + 6 = x³ + 6x² + 11x + 6
Same rules apply!
Example 1: (x + y)(x + 2y)
F: x · x = x² O: x · 2y = 2xy I: y · x = xy L: y · 2y = 2y²
Result: x² + 2xy + xy + 2y² = x² + 3xy + 2y²
Example 2: (2a + b)(3a - 2b)
= 6a² - 4ab + 3ab - 2b² = 6a² - ab - 2b²
Example 3: (x + y)² = x² + 2xy + y²
Forgetting the middle term in perfect squares Wrong: (x + 3)² = x² + 9 Right: (x + 3)² = x² + 6x + 9
Sign errors with negatives Careful: (-3)(-2) = +6, not -6
Not combining like terms After FOIL, always combine!
Forgetting to distribute to ALL terms x(x² + 2x + 1) has THREE terms to multiply
Exponent errors x · x = x², not 2x or x
Example 1: Rectangle with length (x + 5) and width (x + 3). Find area.
Area = length × width = (x + 5)(x + 3) = x² + 3x + 5x + 15 = x² + 8x + 15
Example 2: Square with side length (2x + 1). Find area.
Area = (side)² = (2x + 1)² = 4x² + 4x + 1
Example: Box with dimensions (x + 2), (x + 1), and x. Find volume.
V = length × width × height = (x + 2)(x + 1)(x)
First: (x + 2)(x + 1) = x² + 3x + 2 Then: (x² + 3x + 2)(x) = x³ + 3x² + 2x
Binomial Squared: (a ± b)² = a² ± 2ab + b²
Difference of Squares: (a + b)(a - b) = a² - b²
Sum and Difference: If you see (x + k)(x - k), answer is x² - k²
Trinomial × Monomial: Distribute the monomial to each term
Example: 2x(x + 3)(x - 2)
Method 1 - Left to right: 2x(x + 3) = 2x² + 6x (2x² + 6x)(x - 2) = 2x³ - 4x² + 6x² - 12x = 2x³ + 2x² - 12x
Method 2 - Binomials first: (x + 3)(x - 2) = x² + x - 6 2x(x² + x - 6) = 2x³ + 2x² - 12x
Same answer either way!
Example: The profit from selling x items is (20x - 100) dollars. If you sell (x + 5) batches, what is the total profit?
Total profit = (20x - 100)(x + 5) = 20x² + 100x - 100x - 500 = 20x² - 500
Method 1: Substitute x = 1 Evaluate both expressions with x = 1.
Example: Does (x + 2)(x + 3) = x² + 5x + 6? Left: (1 + 2)(1 + 3) = 3 · 4 = 12 Right: 1² + 5(1) + 6 = 1 + 5 + 6 = 12 ✓
Method 2: Factor your answer If you can factor back to the original, it's correct!
| Product Type | Pattern | Example |
|---|---|---|
| Monomial × Monomial | Multiply coefficients, add exponents | 3x · 5x = 15x² |
| Monomial × Polynomial | Distribute | 2x(x + 3) = 2x² + 6x |
| Binomial × Binomial | FOIL | (x+2)(x+3) = x²+5x+6 |
| (a+b)(a-b) | a² - b² | (x+5)(x-5) = x²-25 |
| (a±b)² | a² ± 2ab + b² | (x+3)² = x²+6x+9 |
Level 1: Monomials
Level 2: Distribute
Level 3: FOIL
Level 4: Special products
Level 5: Complex
Multiply:
Use the distributive property:
Answer:
Multiply using FOIL:
Use FOIL:
F:
Expand:
Use the pattern
Avoid these 3 frequent errors
See how this math is used in the real world
Solve .
Combine:
Answer:
Here and :
Alternative - FOIL:
Result:
Answer: