Multiplying two conjugates, such as (a + b)(a − b), is the most well-known operation that results in a binomial. The middle terms cancel out, leaving exactly two terms: a² − b². But multiplication isn’t the only route. Addition, subtraction, and division can all produce a binomial when like terms cancel or combine in the right way.
A binomial is simply a polynomial with exactly two terms. Expressions like 3x + 7, a² − b², and 4x³ + 5 all qualify. The question of which operation “results in” a binomial comes up most often in algebra courses when studying special products, but it’s worth seeing the full picture.
Multiplying Conjugates: The Classic Case
The most reliable way to get a binomial from multiplication is to multiply a sum by a difference of the same two terms. These pairs are called conjugates: (a + b) and (a − b). When you expand the product using FOIL or distribution, the two middle terms are exact opposites and cancel each other out:
(a + b)(a − b) = a² − ab + ab − b² = a² − b²
The result, a² − b², is called a difference of squares. It has exactly two terms, making it a binomial. This pattern holds no matter what a and b represent. If a = 3x and b = 5, then (3x + 5)(3x − 5) = 9x² − 25. If a = x² and b = 7, then (x² + 7)(x² − 7) = x⁴ − 49. Every time, the cross terms vanish and you’re left with two perfect squares separated by a minus sign.
This is the answer most algebra textbooks are looking for when they ask which operation results in a binomial. It’s a special product formula worth memorizing because it shows up constantly in factoring, simplifying rational expressions, and working with radicals (where you multiply by conjugates to rationalize denominators).
Why Most Multiplication Doesn’t Produce a Binomial
When you multiply two generic binomials like (x + 3)(x + 5), you typically get three terms: x² + 8x + 15. That’s a trinomial, not a binomial. The middle terms only cancel when they’re equal in size and opposite in sign, which is exactly the condition conjugates satisfy. Squaring a binomial, such as (a + b)², gives you a² + 2ab + b², which is also a trinomial. So among all the ways to multiply two binomials together, the conjugate pattern is the specific case that yields a binomial result.
Subtraction That Produces a Binomial
Subtracting one polynomial from another can also leave you with a binomial if enough like terms cancel. Consider this example: (p² + q²) − (p² + 10pq − 2q²). After distributing the negative sign and combining like terms, the p² terms cancel, leaving −10pq + 3q². That’s a binomial.
A more dramatic case: (a³ − a²b) − (ab² + b³) + (a²b + ab²). Once you distribute and rearrange, every pair of middle terms cancels, and you’re left with a³ − b³, a clean binomial known as a difference of cubes. The key principle is straightforward. When you add or subtract polynomials and all but two groups of like terms cancel, the simplified result is a binomial.
Division That Produces a Binomial
Polynomial long division or synthetic division can also yield a binomial quotient. For instance, dividing 4x⁴ + 12x³ + 5x + 9 by x + 3 gives a quotient of 4x³ + 5 (with a remainder of −6). That quotient is a binomial because the x² and x coefficients both work out to zero during the division process.
Similarly, dividing 9x³ − 8x + 13 by 3x² − 4x + 2 produces a quotient of 3x + 4 (with a remainder). There’s no single rule that predicts when a division will produce a binomial, but it happens whenever the intermediate terms in the quotient end up with zero coefficients.
Factoring as the Reverse View
Factoring is the reverse of multiplication, and it frequently produces binomials. When you factor a quadratic trinomial like x² + 7x + 12, you get (x + 3)(x + 4), a product of two binomials. When you factor a difference of squares like 25x² − 9, you get (5x + 3)(5x − 3). A technique called factoring by grouping can break a four-term polynomial into a product of two binomials: for example, 24a⁴ − 18a³ − 20a + 15 factors into (4a − 3)(6a³ − 5).
So factoring doesn’t result in a single binomial. It results in binomial factors. The distinction matters if your textbook is asking which operation yields one binomial expression as its output. In that case, the answer is multiplying conjugates (the difference of squares pattern). But if the question is broader, asking which operations can involve binomials in their results, factoring belongs on the list too.
Putting It Together
If you’re answering a homework or test question, the expected answer is almost always the multiplication of conjugates: (a + b)(a − b) = a² − b². This is the standard special product that reliably produces a binomial. The middle terms always cancel, and the result is always a difference of two squares. Addition, subtraction, and division can also produce binomials in specific situations, but those outcomes depend on the particular expressions involved rather than following a guaranteed pattern.