Polynomial Multiplication

Master FOIL method and polynomial multiplication. Math 10-C Alberta mathematics curriculum.

Lesson 4.2 of Factors and Roots in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Distributive Property and FOIL Review. Master the foundation of polynomial multiplication: every term in the first polynomial multiplies every term in the second. Learn how FOIL is simply a special case of the distributive property, and practice collecting like terms after expanding.
• Binomial Times Trinomial and Squaring Binomials. Extend beyond FOIL to multiply a binomial by a trinomial (6 products), and learn the squaring pattern — why the middle term is always present and how to avoid the most common squaring mistake.
• Multi-Step and Complex Problems. Tackle problems with three factors, subtracted products, and mixed trinomial-binomial expansions. Learn to work in stages, apply the sign-distribution rule carefully, and simplify fully by combining all like terms.

Inside the lesson — a free preview

The Core Rule

The Distributive Property. Every term in the first polynomial must be multiplied by every term in the second polynomial.

For two binomials, this process has a nickname: FOIL (First, Outer, Inner, Last). But FOIL is just a special case of the distributive property — it only works for two binomials.

The general principle works for polynomials of any size.

What Are Like Terms?

Like Terms. Like terms are terms that have the same variables raised to the same powers. Only like terms can be combined after expanding.

Like terms: 3x^2 and -7x^2, or 5xy and 2xy.

Not like terms: 3x^2 and 3x (different exponents), or 5xy and 5x^2y (different exponents).

After multiplying, always collect and combine like terms to simplify.

The Squaring Pattern

Squaring Patterns to Memorize. (a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

Memorizing these patterns saves time. The middle term is always 2ab — positive when adding, negative when subtracting.

Make sure you understand why these work — they come directly from applying FOIL.

Worked example: Binomial Times Binomial (FOIL)

Example 1: (3x - 2y)(4x + y). Multiply every term in the first binomial by every term in the second:

First: (3x)(4x) = 12x^2

Outer: (3x)(y) = 3xy

Inner: (-2y)(4x) = -8xy

Last: (-2y)(y) = -2y^2

Combine like terms (3xy and -8xy):

Answer: 12x^2 - 5xy - 2y^2

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