How to Multiply Polynomials

Multiplying polynomials comes down to one principle: every term in the first polynomial gets multiplied by every term in the second. What changes is how you organize the work as the expressions get larger.

Multiplying Two Binomials — FOIL

When both polynomials have exactly two terms (binomials), there's a shortcut worth knowing called FOIL: First, Outer, Inner, Last. It's just a way to make sure you hit all four pairings without missing one.

For $(x + 3)(x + 5)$:

First: $x \cdot x = x^2$
Outer: $x \cdot 5 = 5x$
Inner: $3 \cdot x = 3x$
Last: $3 \cdot 5 = 15$

Add everything up and combine like terms:

$$x^2 + 5x + 3x + 15 = x^2 + 8x + 15$$

Example 1

Multiply $(2x - 1)(x + 4)$.

$$\underbrace{2x \cdot x}{\text{F}} + \underbrace{2x \cdot 4}{\text{O}} + \underbrace{(-1) \cdot x}{\text{I}} + \underbrace{(-1) \cdot 4}{\text{L}}$$

$$= 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4$$

Watch the signs — the minus in $-1$ carries through to both the Inner and Last terms.

Example 2

Multiply $(3x - 2)(3x + 2)$.

$$= 9x^2 + 6x - 6x - 4 = 9x^2 - 4$$

The middle terms cancelled. This happens whenever you multiply a sum and a difference of the same two terms — a pattern worth recognizing (more on that below).

Larger Polynomials

FOIL only applies to binomials. When one or both polynomials have three or more terms, you need the full distributive method: take each term from the first polynomial and multiply it by the entire second polynomial, then collect everything at the end.

Example 3

Multiply $(x^2 - 2x + 1)(3x^3 + 4x^2 + 5x)$.

Work through each term of the first polynomial:

Distribute $x^2$: $$x^2(3x^3 + 4x^2 + 5x) = 3x^5 + 4x^4 + 5x^3$$

Distribute $-2x$: $$-2x(3x^3 + 4x^2 + 5x) = -6x^4 - 8x^3 - 10x^2$$

Distribute $1$: $$1(3x^3 + 4x^2 + 5x) = 3x^3 + 4x^2 + 5x$$

Now collect all the terms and combine like terms:

$$3x^5 + 4x^4 + 5x^3 - 6x^4 - 8x^3 - 10x^2 + 3x^3 + 4x^2 + 5x$$

Group by degree:

$$= 3x^5 + (4x^4 - 6x^4) + (5x^3 - 8x^3 + 3x^3) + (-10x^2 + 4x^2) + 5x$$

$$= 3x^5 - 2x^4 + 0x^3 - 6x^2 + 5x$$

$$= 3x^5 - 2x^4 - 6x^2 + 5x$$

The $x^3$ terms cancelled entirely, which is fine — always double-check by accounting for every term from each step.

Example 4

Multiply $(x + 2)(x^2 - 3x + 4)$.

Distribute $x$: $$x(x^2 - 3x + 4) = x^3 - 3x^2 + 4x$$

Distribute $2$: $$2(x^2 - 3x + 4) = 2x^2 - 6x + 8$$

Combine: $$x^3 - 3x^2 + 4x + 2x^2 - 6x + 8 = x^3 - x^2 - 2x + 8$$

Special Products

A few patterns come up so often they're worth memorizing:

Sum and difference (difference of squares): $$\boxed{(a + b)(a - b) = a^2 - b^2}$$

The middle terms always cancel. Example: $(x+5)(x-5) = x^2 - 25$.

Perfect square trinomials: $$\boxed{(a + b)^2 = a^2 + 2ab + b^2}$$ $$\boxed{(a - b)^2 = a^2 - 2ab + b^2}$$

A common mistake is writing $(x+3)^2 = x^2 + 9$. That's wrong — the middle term $2(x)(3) = 6x$ gets dropped. The correct answer is $x^2 + 6x + 9$.

Cube of a binomial: $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$

Practice Problems

1. Multiply $(x + 6)(x - 2)$. Show answerFOIL: $x^2 - 2x + 6x - 12 = x^2 + 4x - 12$.

2. Multiply $(2x + 3)^2$. Show answerPerfect square: $(2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$.

3. Multiply $(x - 4)(x + 4)$. Show answerSum and difference: $x^2 - 16$. The middle terms cancel.

4. Multiply $(x + 1)(x^2 + 2x - 3)$. Show answerDistribute $x$: $x^3 + 2x^2 - 3x$. Distribute $1$: $x^2 + 2x - 3$. Combine: $x^3 + 3x^2 - x - 3$.

5. Multiply $(2x^2 - x)(x^2 + 3x - 1)$. Show answerDistribute $2x^2$: $2x^4 + 6x^3 - 2x^2$. Distribute $-x$: $-x^3 - 3x^2 + x$. Combine: $2x^4 + 5x^3 - 5x^2 + x$.