# Multiplying Binomials Calculator

Created by Maciej Kowalski, PhD candidate
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

Welcome to Omni's multiplying binomials calculator, where we'll learn just what the name suggests: how to multiply binomials. It is a special case of polynomial multiplication, which we covered in the multiplying polynomials calculator, but it's so common in applications and coursebooks that it deserved its very own calculator. So for today, binomials are the only polynomials we need (well, sort of...), and if you're not too sure what they are, we give the binomial definition in math just in case.

## Binomials, polynomials

They call mathematics the language of the universe for a reason: it describes the rules that govern the world. Sure, physicists do something similar, but in the end, physics is just applying mathematics to specific scenarios. *grabs popcorn*

Anyway, the way we present those rules is through formulas. These vary in length and difficulty from that for the area of a rectangle, through the mass moment of inertia, to some crazy equations only a handful of people understand (or claim to understand). All of these have one important thing in common: variables.

Variables represent objects (usually numbers) that we don't know or don't want to specify. For instance, the formula for gravitational pull involves the variable $m$ for mass, but since we don't give the number straight away, we can use the equation for the Earth, the Moon, or any other object. The variables make the whole thing universal.

Polynomials are expressions that contain variables only in non-negative integer powers. In other words, a polynomial cannot involve variables inside roots, logarithms, trigonometric functions, or any other fancy mathematical tool. However, they can involve many variables.

Below, we list a few examples of polynomials:

• $x + 2y$

• $a^2 + 2ab + b$

• $n^3 - 0.7n + \frac{3}{8}$

• $1 + 3 + x^5 - x^7 + 19x^9$

Note how the exponents can differ; they can even repeat, as long as they aren't negative or fractional. For the latter, you can refer to our fraction exponent calculator.

A binomial is a polynomial with two terms. As such, among the ones above, only the first expression is a binomial. However, note that although the binomial definition in math is fairly general, Omni's multiplying binomials calculator deals only with so-called linear binomials. To be precise, we'll have expressions of the form $ax + b$, meaning with only one variable $x$ and in the first power. Such binomials are most common in applications and coursebooks and are more than enough to explain the concept.

Now that we've come to know our enemy, we're ready to fight them! Let's learn how to multiply binomials.

## How to multiply binomials?

For those who like word-based explanations, there is a method of multiplying binomials called the FOIL method. If you're interested, feel free to check out Omni's FOIL calculator. Here, however, we focus on formulas.

When multiplying binomials (or any polynomials, for that matter), the basic rule is: multiply every term of the first expression by every term of the second. We don't want to go into too much generality, so let's just take two binomials (for simplicity, we'll use the same notation our multiplying binomials calculator uses): $a_1x + a_0$ and $b_1x + b_0$. If we apply the rule mentioned, we get:

$(a_1x + a_0)(b_1x + b_0) \\[1em] = (a_1x\times b_1x) + (a_1x \times b_0) \\[1em] \ \ \ \ + (a_0 \times b_1x) + (a_0 \times b_0)$

Recall that $x \times x = x^2$ and that we can take the numbers in front of the variables (because multiplication is commutative). Therefore:

$(a_1x + a_0)(b_1x + b_0) \\[1em] = (a_1x\times b_1x) + (a_1x \times b_0) \\[1em] \ \ \ \ + (a_0 \times b_1x) + (a_0 \times b_0) \\[1em] = (a_1\times b_1)x^2 + (a_1\times b_0)x \\[1em] \ \ \ \ (a_0\times b_1)x + (a_0 \times b_0)$

Next, we group together the two summands with $x$ (i.e., use the fact that multiplication is associative), and get:

$(a_1x + a_0)(b_1x + b_0) \\[1em] = (a_1b_1)x^2 + (a_1b_0)x \\[1em] \ \ \ \ + (a_0b_1)x + (a_0b_0) \\[1em] = (a_1b_1)x^2 + (a_1b_0 + a_0b_1)x \\[1em] \ \ \ \ + (a_0b_0)$

VoilΓ ! We got the formula for multiplying binomials. Let us just mention here that it's not too difficult to go from binomials to polynomials in general; there's just a bit more work to do.

If you've multiplied binomials and want to learn how to undo this operation, check our factoring trinomials calculator.

We've seen the binomial definition in math, we've learned how to multiply binomials, so there's only one thing left to do: see an example.

## Example: using the multiplying binomials calculator

Let's find the product of $3x - 2$ and $x + 5$. Before we get our hands dirty (well, not really), we'll show you how to get the answer using the multiplying binomials calculator.

At the top of our tool, we see a symbolic expression representing the problem at hand:

$(a_1x + a_0)(b_1 x + b_0) \\[1em] = c_2 x^2 + c_1 x + c_0$

The same notation is used underneath in corresponding sections. Looking back at our example, we input:

$a_1 = 3, a_0 = -2, b_1 = 1, b_0 = 5$

Note how we have $b_1 = 1$ even though there was no $1$ in our binomial. That is because by convention, if the number in front of the variable is $1$, we don't write it. Also, observe how some c's get calculated even before we input all four entries. For instance, it's enough to have $a_1$ and $b_1$ to see what $c_2$ is because the formula in the above section needs no other values. Lastly, note how the multiplying binomials calculator also gives a step-by-step solution once you give it all the necessary numbers.

But before we reveal the answer, let's try to arrive at it ourselves. The task shouldn't be too difficult: we simply apply the formula from the above section and do the simple addition and multiplication.

$(3x -2)(x + 5) \\[1em] = (3\times1)x^2 \\[1em] \ \ + (3\times5 + (-2) \times 1)x \\[1em] \ \ + ((-2)\times5) \\[1em] = 3x^2 + 13x - 10$

And just like that, we're done! A piece of cake, wouldn't you say?

Make sure to check out other Omni tools dedicated to polynomials in the algebra calculators section, for example, the polynomial division calculator.

Maciej Kowalski, PhD candidate
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Second binomial
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