Multiplying Polynomials: FOIL, Box Method, Made Easy

Multiplying polynomials uses one rule repeatedly: every term in the first polynomial multiplies every term in the second. For (x + 3)(x + 5), x multiplies both x and 5, then 3 multiplies both x and 5, giving x² + 5x + 3x + 15, which simplifies to x² + 8x + 15.

This skill powers the famous algebraic identities, quadratic equations and factorisation. Whenever a rectangle's sides are expressions, a profit formula multiplies price by quantity, or a physicist squares a binomial, polynomial multiplication is doing the work.

Key Concepts

• Distributive law: a(b + c) = ab + ac — the single rule behind all polynomial multiplication.
• FOIL: for two binomials, multiply First, Outer, Inner, Last terms: (a + b)(c + d) = ac + ad + bc + bd.
• Box (grid) method: draw a table with one polynomial's terms along the top and the other's down the side; each cell is a product, then add all cells.
• Degree of product: degrees add — a degree-2 times a degree-3 polynomial gives degree 5.
• Exponent rule: xᵐ × xⁿ = xᵐ⁺ⁿ (add exponents when multiplying same bases).

Formulas

• Monomial × polynomial: 2x(3x² − x + 4) = 6x³ − 2x² + 8x.
• Binomial × binomial (FOIL): (a + b)(c + d) = ac + ad + bc + bd.
• Binomial × trinomial: multiply each of the 2 terms by each of the 3 terms (6 products), then combine like terms.
• Assumes standard arithmetic on coefficients and the exponent rule for variables.

Worked Example

Multiply: (2x − 1)(x + 4).

1. FOIL — First: 2x × x = 2x².
2. Outer: 2x × 4 = 8x. Inner: −1 × x = −x. Last: −1 × 4 = −4.
3. Combine the middle terms: 8x − x = 7x.

The product is 2x² + 7x − 4. Verify with x = 2: (4 − 1)(2 + 4) = 3 × 6 = 18; the product gives 2(4) + 14 − 4 = 8 + 14 − 4 = 18. Matches.

Common Misconceptions

• "(x + 3)² = x² + 9." Wrong — the middle term is missing. (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9. The cross terms always double.
• "Multiply the exponents when multiplying." Exponents add: x² × x³ = x⁵, not x⁶. (Exponents multiply for powers of powers, like (x²)³ = x⁶.)
• "Signs can be fixed at the end." No — a dropped minus in one of six products corrupts the whole answer. Attach each term's sign before multiplying.

Real-World Applications

• Finding areas where length and width are algebraic expressions.
• Expanding the identities used in mental-math shortcuts.
• Physics formulas that multiply binomial quantities, like (v + u)(v − u).
• Computer algebra systems multiply polynomials in cryptography and coding theory.

Polynomial multiplication is the distributive law applied thoroughly: every term times every term, signs attached, then combine like terms. FOIL handles binomials, the box method organises bigger products, and a quick substitution check catches errors.