Multiplying Polynomials: The Box Method and FOIL

Every term meets every term

Multiplying polynomials is one rule applied repeatedly: the distributive property.

$$a(b + c) = ab + ac$$

With a single term outside, it is one pass:

$$3x(x^2 - 4x + 5) = 3x^3 - 12x^2 + 15x$$

Each term inside was multiplied by $3x$. Nothing outside the parentheses was left undistributed, and nothing inside was skipped.

When both sets of parentheses hold several terms, the same thing happens on a larger scale. Every term of the first set of parentheses must meet every term of the second. That is the whole job, and the only difficulty is keeping track.

The box method

Draw a box with one row per term of the first polynomial and one column per term of the second. Fill each cell with the product of its row and column.

$$(x + 3)(x + 5)$$

$\times$	$x$	$+5$
$x$	$x^2$	$5x$
$+3$	$3x$	$15$

Add the cells and collect like terms:

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

The box is worth using because the number of cells is fixed before you start. Two terms times two terms is four cells; if you have not filled four cells, you have not finished.

This is the same picture as splitting a rectangle to find its area, with letters where the numbers were. A rectangle $x + 3$ by $x + 5$ divides into four smaller rectangles, and its area is their total.

FOIL is the box, named

For two binomials, the four cells have traditional names:

Letter	Means	In $(x + 3)(x + 5)$
F	First terms	$x \cdot x = x^2$
O	Outer terms	$x \cdot 5 = 5x$
I	Inner terms	$3 \cdot x = 3x$
L	Last terms	$3 \cdot 5 = 15$

Same four products, same answer. FOIL is a way of remembering the order.

Its limit is built in. FOIL names exactly four products, so it fits exactly the $2 \times 2$ case. A binomial times a trinomial has six products, and the name runs out after four.

The box does not have that problem, so it is the one worth making a habit.

Bigger products

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

Two rows, three columns, six cells:

$\times$	$x^2$	$-3x$	$+4$
$x$	$x^3$	$-3x^2$	$4x$
$+2$	$2x^2$	$-6x$	$8$

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

Collect down the diagonals — cells on the same diagonal hold the same power, which is why the box lines them up for you.

Why the degree adds

Multiplying $x^m$ by $x^n$ gives $x^{m+n}$, so the highest power of the product comes from the two leading terms multiplied together.

$$\deg(fg) = \deg f + \deg g$$

A quadratic times a cubic is degree $5$. Nothing else in the box can reach that high, and the two leading coefficients cannot cancel each other, so the leading term always survives.

This is a fast check. If your answer to $(x+2)(x^2-3x+4)$ has no $x^3$ term, something was dropped.

Two products worth knowing on sight

Difference of squares.

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

The box shows why the middle vanishes: the outer product is $+ab$ and the inner is $-ab$, so they cancel.

$$(x - 7)(x + 7) = x^2 - 49$$

Perfect square trinomial.

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

Squaring means multiplying the binomial by itself, so the outer and inner products are both $ab$ — and two of them make $2ab$.

$$(x + 5)^2 = x^2 + 10x + 25$$

$(a+b)^2$ is not $a^2 + b^2$. Test it with numbers: $(3+4)^2 = 49$, while $9 + 16 = 25$. The missing $24$ is the $2ab$.

Recognizing these two saves work now and is what makes factoring fast later, because factoring is this read backwards.

Worked examples

Common mistakes

Practice problems