7.3Multiplying Polynomials
Multiply polynomials using the distributive property and the FOIL method. Master monomial × polynomial, binomial × binomial, and polynomial × polynomial — the foundation for factoring and special products.
Why This Matters
Multiplying polynomials — especially FOIL and the distributive property — is used constantly in Algebra 2, Precalculus, and Calculus. It's also the foundation for understanding how quadratic expressions are formed.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do we multiply polynomials, and why must every term in one factor be multiplied by every term in the other?
Lesson Overview
Multiplying polynomials is an extension of the distributive property: every term in the first factor must be multiplied by every term in the second. For monomial × polynomial, distribute the monomial to each term. For binomial × binomial, use FOIL (First, Outer, Inner, Last) or the box method to track all four partial products. For larger polynomials, distribute each term of the first polynomial to every term of the second, then collect and combine like terms. Always write the final answer in standard form and verify the degree equals the sum of the factors' degrees.
Exponent Rules — Quick Reference for Multiplication
Product Rule
xᵃ · xᵇ = xᵃ⁺ᵇ
e.g. x³ · x² = x⁵
Add exponents — same base
Power Rule
(xᵃ)ᵇ = xᵃᵇ
e.g. (x²)³ = x⁶
Multiply exponents
Coeff × Coeff
(aˣ)(bˣ) = (ab)ˣ
e.g. 3x² · 4x = 12x³
Multiply coefficients, add exponents
Distributive Property — Monomial × Polynomial
3x(2x² − 5x + 4)
= 6x³ − 15x² + 12x
−2x²(x³ + 3x − 1)
= −2x⁵ − 6x³ + 2x²
FOIL Method — (a + b)(c + d)
Example: (x + 3)(x + 5)
Combine like terms (5x + 3x):
x² + 8x + 15
FOIL Arc Diagram
x·x
x·5
3·x
3·5
Box (Area) Method — (2x + 3)(x + 4)
Step 1 — Fill the grid
| × | x | +4 |
|---|---|---|
| 2x | 2x² | 8x |
| +3 | 3x | 12 |
Step 2 — Add all cells
2x² + 8x + 3x + 12
Combine like terms: 8x + 3x = 11x
= 2x² + 11x + 12
Why the box method works:
Each cell is one partial product. The box ensures every term in the first factor is multiplied by every term in the second — no terms are missed.
FOIL vs Box Method — Same Problem, Two Approaches
Problem: (3x − 2)(2x + 5)
FOIL Method
F: 3x · 2x = 6x²
O: 3x · 5 = 15x
I: −2 · 2x = −4x
L: −2 · 5 = −10
6x² + 15x − 4x − 10
= 6x² + 11x − 10
Box Method
| × | 2x | +5 |
|---|---|---|
| 3x | 6x² | 15x |
| −2 | −4x | −10 |
= 6x² + 11x − 10
Polynomial Multiplication — Decision Flowchart
What type of multiplication?
Monomial × Monomial
Multiply coefficients, add exponents
3x² · 4x = 12x³
Monomial × Polynomial
Distribute to each term
2x(x+3) = 2x²+6x
Polynomial × Polynomial
FOIL (binomials) or Box/Distribute
(x+1)(x+2) = x²+3x+2
Worked Examples
Multiply monomials: (3x²)(4x³)
Multiply coefficients: 3 × 4 = 12
Add exponents (same base x): 2 + 3 = 5
Result: 12x⁵
Monomial × Polynomial: 3x(2x² − 5x + 4)
Distribute 3x to each term:
3x · 2x² = 6x³
3x · (−5x) = −15x²
3x · 4 = 12x
FOIL: (x + 3)(x + 5)
F (First): x · x = x²
O (Outer): x · 5 = 5x
I (Inner): 3 · x = 3x
L (Last): 3 · 5 = 15
Sum: x² + 5x + 3x + 15
Combine like terms: x² + 8x + 15
Box Method: (2x + 3)(x + 4)
Set up a 2×2 grid with 2x and +3 on one side, x and +4 on the other.
Fill cells: 2x·x = 2x², 2x·4 = 8x, 3·x = 3x, 3·4 = 12
Sum all cells: 2x² + 8x + 3x + 12
Combine like terms: 2x² + 11x + 12
FOIL with negatives: (3x − 2)(2x + 5)
F: 3x · 2x = 6x²
O: 3x · 5 = 15x
I: −2 · 2x = −4x
L: −2 · 5 = −10
Sum: 6x² + 15x − 4x − 10
Combine: 6x² + 11x − 10
Guided Practice
Guided Practice Video: Multiplying Polynomials
Review FOIL and polynomial multiplication using distribution before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Multiply: 5x²(2x³ − 3x + 1)
Hint: Distribute 5x² to each term. Multiply coefficients and add exponents for each term.
FOIL: (x + 6)(x − 2)
Hint: F: x·x; O: x·(−2); I: 6·x; L: 6·(−2). Then combine the two middle terms.
Box method: (3x − 1)(2x + 5)
Hint: Set up a 2×2 grid. Fill each cell with one partial product. Sum all four cells, then combine like terms.
Multiply: (x + 3)(x² − 2x + 4)
Hint: Distribute x to all three terms of the trinomial, then distribute 3 to all three terms. Sum all six partial products.
A square has side length (x + 7). Write and simplify an expression for its area.
Hint: Area = side². Write (x + 7)(x + 7) and apply FOIL.
Key Vocabulary
FOIL Method
A mnemonic for multiplying two binomials: First, Outer, Inner, Last. Each letter represents one partial product.
Distributive Property
a(b + c) = ab + ac. The foundation of all polynomial multiplication.
Partial Product
One of the individual products formed when multiplying polynomials. All partial products are summed to get the final answer.
Product Rule
xᵃ · xᵇ = xᵃ⁺ᵇ. When multiplying powers with the same base, add the exponents.
Box Method
A grid-based method for multiplying polynomials. Each cell holds one partial product; all cells are summed.
Degree of a Product
The degree of a product equals the sum of the degrees of the factors.
Leading Term
The term with the highest degree in a polynomial. In FOIL, it comes from First × First.
Constant Term
The term with degree 0 (no variable). In FOIL, it comes from Last × Last.
Interactive Practice — 5 Questions
What is (3x²)(4x³)?
What is the result of 2x(x² + 3x − 5)?
Using FOIL, what is (x + 4)(x + 3)?
What is (x − 5)(x + 5)?
What is the degree of the product (x² + 1)(x³ − x)?
Independent Practice
Independent Practice
Multiply using the distributive property: (a) 4x(x² − 3x + 2) (b) −2x²(3x³ + x − 5). Show each partial product.
FOIL: (a) (x + 5)(x − 3) (b) (2x + 1)(3x − 4). Label each step F, O, I, L and combine like terms.
Box method: (a) (x + 6)(x + 2) (b) (3x − 2)(2x + 5). Draw the grid, fill each cell, then sum.
Multiply a trinomial: (a) (x + 2)(x² − x + 3) (b) (2x − 1)(x² + 4x − 2). Distribute each term of the first factor.
Real-world: (a) A rectangle has length (3x + 1) and width (x + 5). Find the area. (b) A square has side (x + 4). Find the area. Write both answers in standard form.
Common Mistakes
Multiplying only the first terms of each binomial — e.g., (x + 3)(x + 5) = x² + 15.
Use FOIL or the distributive property to multiply every term by every other term: x² + 5x + 3x + 15 = x² + 8x + 15.
Adding exponents when multiplying coefficients — e.g., 3x · 4x = 7x².
Multiply the coefficients and add the exponents: 3x · 4x = 12x².
Forgetting the middle term when squaring a binomial — e.g., (x + 4)² = x² + 16.
(x + 4)² = x² + 8x + 16. The middle term 2(x)(4) = 8x is always present.
Combining unlike terms in the product — e.g., writing 2x³ + 3x² as 5x⁵.
Only combine like terms (same variable, same exponent). 2x³ and 3x² are unlike terms.
Math Tips
Monomial × Monomial: multiply coefficients, add exponents.
Monomial × Polynomial: distribute to every term — don't skip any.
FOIL: First, Outer, Inner, Last — four products, then combine the middle two.
Box method: great for larger polynomials — every cell is one partial product.
Signs: negative × negative = positive. Track signs carefully in each partial product.
Degree check: degree of answer = sum of degrees of factors. Use this to catch errors.
Always write the final answer in standard form (highest degree first).