Multiplying Binomials Calculator
Welcome to CalcuGo'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 cover in the multiplying polynomials calculator, but it's so common in applications and coursebooks that it deserves its very own calculator. So for today, binomials are the only polynomials we need — 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.
Anyway, the way we present those rules is through formulas. These vary in length and difficulty from the one 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 them 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 + 2ya² + 2ab + bn³ − 0.7n + 3/81 + 3 + x⁵ − x⁷ + 19x⁹
Note how the exponents can differ; they can even repeat, as long as they aren't negative or fractional.
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, this 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 CalcuGo'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): a₁x + a₀ and b₁x + b₀. If we apply the rule mentioned, we get:
(a₁x + a₀)(b₁x + b₀) = (a₁x × b₁x) + (a₁x × b₀) + (a₀ × b₁x) + (a₀ × b₀)
Recall that x × x = x² and that we can take the numbers in front of the variables (because multiplication is commutative). Therefore:
(a₁x + a₀)(b₁x + b₀) = (a₁ × b₁)x² + (a₁ × b₀)x + (a₀ × b₁)x + (a₀ × b₀)
Next, we group together the two summands with x (i.e., use the fact that multiplication is distributive over addition), and get the formula for multiplying binomials:
(a₁x + a₀)(b₁x + b₀) = (a₁b₁)x² + (a₁b₀ + a₀b₁)x + a₀b₀
Observe that the result is always a quadratic (degree-two) polynomial, as long as a₁ and b₁ are non-zero. Writing the product as c₂x² + c₁x + c₀, the three coefficients are:
c₂ = a₁ × b₁— the leading coefficient;c₁ = a₁ × b₀ + a₀ × b₁— the middle coefficient, and the only one that needs two products;c₀ = a₀ × b₀— the constant term.
The box method
The calculator above shows the four partial products arranged in a 2 × 2 box (also called the area model or grid method). Write the terms of the first binomial across the top, the terms of the second binomial down the side, and fill each cell with the product of its row and column headers. Adding up all four cells gives the answer.
The box method is worth knowing because, unlike FOIL, it scales: to multiply a trinomial by a binomial you simply draw a 3 × 2 box instead. FOIL's four letters only ever describe the binomial-times-binomial case.
Example: using the multiplying binomials calculator
Say we want to multiply (2x + 3) and (x − 4). Matching this against a₁x + a₀ and b₁x + b₀ gives a₁ = 2, a₀ = 3, b₁ = 1, and b₀ = −4. Note the minus sign on b₀: the subtraction in x − 4 means the constant term is negative four, not four.
Now apply the formula:
c₂ = 2 × 1 = 2c₁ = 2 × (−4) + 3 × 1 = −8 + 3 = −5c₀ = 3 × (−4) = −12
So (2x + 3)(x − 4) = 2x² − 5x − 12. Enter the four coefficients above and the calculator reports exactly this, along with the box, the four partial products, and the sum before the like terms are combined.
How do you use this multiplying binomials calculator?
- Write each factor in the form
a₁x + a₀. Terms that are subtracted get a negative coefficient. - Enter
a₁anda₀for the first binomial, thenb₁andb₀for the second. Watch the live preview to confirm you typed what you meant. - Optionally enter a value of
xto evaluate the product numerically, and pick a unit or currency for that number. - Read off the box, the coefficient table, the roots, and the simplified answer.
The leading coefficients a₁ and b₁ must be non-zero — otherwise the factor has no x term and isn't a linear binomial at all. If you set a₀ or b₀ to zero, the calculator still multiplies correctly but warns you that the factor has collapsed to a monomial.
The roots come for free
Here's a bonus that the expanded form hides. A product equals zero exactly when one of its factors is zero, so the roots of (a₁x + a₀)(b₁x + b₀) can be read straight off the factors:
x₁ = −a₀ / a₁ and x₂ = −b₀ / b₁
No quadratic formula required — the factored form is the solution. For our example, x₁ = −3/2 and x₂ = 4.
There's an elegant consequence. The discriminant of the product works out to:
Δ = c₁² − 4c₂c₀ = (a₁b₀ + a₀b₁)² − 4a₁b₁a₀b₀ = (a₁b₀ − a₀b₁)²
which is a perfect square, and therefore never negative. So the product of two linear binomials with real coefficients always has two real roots — it can never be an irreducible quadratic like x² + 1. And Δ = 0 precisely when a₁b₀ = a₀b₁, i.e. when the two binomials are proportional and the two roots coincide.
Special products worth memorizing
Two combinations show up often enough to be worth recognizing on sight. The calculator flags them automatically:
- Difference of squares:
(a + b)(a − b) = a² − b². The middle terms cancel, soc₁ = 0. Try(5x − 1)(5x + 1) = 25x² − 1. - Perfect square:
(a + b)² = a² + 2ab + b²and(a − b)² = a² − 2ab + b². Try(x + 3)² = x² + 6x + 9.
Going the other way: factoring
Reading the formula right to left turns multiplication into factoring. To factor x² + Bx + C, look for two numbers p and q with p × q = C and p + q = B; then x² + Bx + C = (x + p)(x + q).
Example: factor x² + x − 6. We need p × q = −6 and p + q = 1, so p = 3 and q = −2, giving (x + 3)(x − 2). Run those coefficients through the calculator to verify the factorization instantly — multiplying back is the fastest check there is.
Units and currencies
Multiplying binomials is a symbolic operation — the product is a polynomial, not a measured quantity, so the polynomial itself carries no units. That's why the unit selector applies to the optional evaluation step: when you supply a value of x, the calculator computes the numeric value of the product and labels it with your chosen unit.
This matters for real problems. A rectangle whose sides measure (2x + 3) and (x − 4) feet has area 2x² − 5x − 12 square feet; a price of (x + 5) dollars on (3x − 2) units gives revenue 3x² + 13x − 10 dollars. Pick a world currency (US dollar, Russian ruble, euro, pound, yen and 15 more), a metric unit (meters, kilograms, liters, °C), or a US/imperial unit (feet, pounds, gallons, °F) to label the evaluated result.