Multiplying Polynomials

Multiplying one term by one term

Before multiplying whole polynomials, start with the smallest piece: a single term times a single term. A term is a coefficient times a power of $x$, so a product of two terms is a product of two numbers and two powers. Multiplication lets you rearrange those factors into any order. Group the numbers together and the powers together:

The coefficients multiply the ordinary way, $3 \cdot 4 = 12$. The powers combine by the product rule for exponents, which says $x^m \cdot x^n = x^{m+n}$, so you add the exponents:

Multiply the coefficients, add the exponents. You add rather than multiply the exponents because $x^2 \cdot x^3$ is two $x$‘s times three more $x$‘s. That is five $x$‘s multiplied in a row, not six. When a term carries a negative sign, fold the sign into the coefficient and proceed exactly the same way:

That is the one small skill every product in this lesson is built from. Everything else is doing it many times and adding up the pieces.

Multiplying a polynomial by a single term

When one factor has several terms, the distributive property takes over. It says that a single factor multiplies each term of a sum separately:

The lone factor $a$ reaches every term inside the parentheses, not just the first. When $a$ and each of $b, c, d$ are themselves terms in $x$, every one of those small products is a one-term-times-one-term computation you just practiced.

Multiplying two binomials

Now both factors are sums, and the distributive property still does all the work. Look at $(x + 2)(x + 3)$. If $x$ is a positive length, then $x + 2$ and $x + 3$ are side lengths, and the product is the area of a rectangle with those sides. Splitting each side at its two parts cuts the rectangle into four smaller rectangles, and their four areas are exactly the four partial products.

The two cross rectangles, $3x$ on the top right and $2x$ on the bottom left, are the Outer and Inner products. They carry the same power of $x$, so they combine into the middle term, and this is the pair that is so easy to forget.

The same thing happens for any two binomials, not just this one. Hold the second binomial $(c + d)$ together as a single block for a moment: $(a + b)$ distributes across it, and each of the two pieces distributes again.

Those four products have a well-known nickname. Reading them in the order First, Outer, Inner, Last spells FOIL: the product of the First terms ($ac$), the Outer terms ($ad$), the Inner terms ($bc$), and the Last terms ($bd$). FOIL is a handy label, but it only counts to four. It says nothing about a binomial times a trinomial, where there are six products instead. The rule that always works, no matter how many terms each factor has, is the plain idea behind FOIL: each term times each term.

Negative terms bring nothing new. Keep each sign attached to its term, and let the sign ride through every product.

The general rule: every term times every term

Nothing about the method depended on each factor having exactly two terms. The same distribution works for any two polynomials, and it reads as one sentence:

To multiply two polynomials, multiply every term of the first by every term of the second, then combine like terms and write the result in standard form.

A factor with $m$ terms times a factor with $n$ terms produces $m \times n$ products before you combine anything. A binomial times a binomial gives $2 \times 2 = 4$ products, which is the case FOIL covers; a binomial times a trinomial gives $2 \times 3 = 6$.

The special products from the factorizations chapter are this same rule, just with the like terms already collected. Squaring $(a + b)(a + b)$ gives four products that collapse to $a^2 + 2ab + b^2$, and the difference of squares $(a + b)(a - b)$ has its two cross terms cancel to leave $a^2 - b^2$. Neither is a separate fact to memorize; this lesson keeps the method general instead.

With more than four products to track, a grid keeps every one of them in its own cell so none can be lost. Put the terms of one factor along the top and the terms of the other down the side. Then fill each cell with the product of its row and column headers.

The degree of a product

Adding polynomials could sometimes lower the degree, when equal leading terms canceled. For two nonzero polynomials, multiplying them never springs that surprise. Multiply the two leading terms, $ax^m$ from one polynomial and $bx^n$ from the other, and you get $ab\,x^{m+n}$. Every other partial product pairs a lower-degree term from one factor with a term from the other, so its power of $x$ is strictly below $m + n$, nothing left to cancel the top term. And $ab$ itself cannot be zero, because $a$ and $b$ are leading coefficients, and a leading coefficient is never zero. So the degree of the product is always $m + n$, the sum of the two degrees. Multiplication cannot lose its leading term the way addition sometimes can. One more fact falls out for free: multiplying two polynomials always gives another polynomial, never anything of a different shape.

Common mistakes