4 of the 8 products merged into others when like terms were collected, leaving 4 terms.
Degree
3
the sum of the 3 bracket degrees
Special shape
roots → polynomial
every bracket is (x − r)
Built up one bracket at a time
The running product after each bracket is multiplied in
After multiplying
Running product
Terms
Degree
(x + 1)(x + 2)
x² + 3x + 2
3
2
(x + 1)(x + 2)(x + 3)
x³ + 6x² + 11x + 6
4
3
3 binomials give 2³ = 8 products, because each bracket contributes one of its two terms and the choices are independent. Collecting like terms brings those 8 down to 4. The count is the check: if you cannot account for all of them before collecting, one has been dropped.
The roots, and the coefficients they produce
Each elementary symmetric function of the roots and the coefficient it becomes
Sum of all products of k roots
Becomes the coefficient of
e1
−6
6 · x2
e2
11
11 · x
e3
−6
6 · x⁰ (the constant)
Roots: −1, −2, −3. Every bracket is (x − r), so this product is the polynomial with roots −1, −2, −3. Its coefficients are the elementary symmetric functions of those roots with alternating signs — e₁ = −6 is their sum, e₃ = −6 is their product — which is Vieta's theorem read forwards. It also means this page answers the reverse of a root-finder: give it the roots and it gives the polynomial.
FOIL names four products because 2² = 4. With 3 brackets there are 8, and no acronym covers them — but the counting argument is unchanged.
The degree of the answer is 3, which is the sum of the 3bracket degrees — a second check that nothing was lost.
Brackets with any other number of terms belong on the Multiplying Polynomials Calculator, where the 2ⁿ count does not apply.
What this tool shows
n brackets, 2ⁿ products, and a count that keeps working where the acronym runs out.
The 2ⁿ product count, and how many survive collecting
The build-up one bracket at a time, with the degree at each stage
The roots reading: (x − r₁)…(x − rₙ) and its symmetric functions
Pascal’s row when every bracket is identical
Two to six brackets 2ⁿ products as the check Roots → polynomial Free, no signup
Updated 7 September 2026 · Works in any browser, no installation
n binomials give exactly 2ⁿ products, because each bracket contributes one of its two terms and the choices are independent. (x + 1)(x + 2)(x + 3) gives 8 products that collect to 4 terms: x³ + 6x² + 11x + 6.
At a glance
Formula shown
n binomials give 2\u207f products before collecting. When every bracket is (x \u2212 r), the coefficients are the elementary symmetric functions of the roots with alternating signs.
Scenario support
Two to six brackets of exactly two terms each, in any number of variables.
Educational estimate
Planning support from the values you enter — not professional advice.
Why 2ⁿ, and why FOIL has exactly four letters
Building one product means walking through the brackets and picking one of the two terms from each. Those picks are independent, so there are 2 × 2 × … × 2 of them — 2ⁿ.
At n = 2 that is 4, which is precisely why FOIL has four letters. First, Outer, Inner and Last are not four arbitrary rules; they are the four ways to pick one term from each of two brackets. (2x + 3)(x − 5) = 2x² − 7x − 15, from 4 products.
3 binomials give 2³ = 8 products, because each bracket contributes one of its two terms and the choices are independent. Collecting like terms brings those 8 down to 4. The count is the check: if you cannot account for all of them before collecting, one has been dropped.
Past two brackets the mnemonic stops and the count does not
(x + 1)(x + 2)(x + 3) has 8 products. There is no letter for the fifth through eighth, and no extended acronym worth learning — but nothing about the method has changed.
The practical approach is to multiply two brackets, collect, then bring in the next. The tool shows that build-up because it is what you actually do, and because each intermediate is itself checkable: (x + 1)(x + 2) = x² + 3x + 2; then (x + 1)(x + 2)(x + 3) = x³ + 6x² + 11x + 6.
Collecting at each stage rather than at the end is worth the habit. It keeps the running product short, which means fewer products at the next stage — and fewer chances to drop one.
Roots in, polynomial out
Here is the reading that makes this a different tool rather than a longer FOIL. When every bracket is (x − r), the product is the polynomial whose roots are those r values — and its coefficients are not arbitrary.
(x − 1)(x − 2)(x − 3)(x − 4) = x⁴ − 10x³ + 35x² − 50x + 24. Its roots are 1, 2, 3, 4, and look at where the coefficients come from: e1 = 10, e2 = 35, e3 = 50, e4 = 24 — the sum of the roots, the sum of their pairwise products, the sum of their triple products, and their product.
Every bracket is (x − r), so this product is the polynomial with roots 1, 2, 3, 4. Its coefficients are the elementary symmetric functions of those roots with alternating signs — e₁ = 10 is their sum, e₄ = 24 is their product — which is Vieta's theorem read forwards. It also means this page answers the reverse of a root-finder: give it the roots and it gives the polynomial.
That is Vieta’s theorem, which the Cubic Equation Calculator uses in the other direction as a check on roots it has found. Used forwards it answers a question no root-finder can: given the roots, what polynomial has them?
When every bracket is the same
(x + 2)(x + 2)(x + 2) is (x + 2)³, and its coefficients are 1, 3, 3, 1 — a row of Pascal’s triangle.
All 3 brackets are the same, so this is (x + 2)³ and the binomial theorem applies: the coefficients are row 3 of Pascal's triangle, 1, 3, 3, 1. That is not a coincidence — the coefficient of a term counts how many of the n brackets contributed which piece, which is exactly what a binomial coefficient counts.
(a + b)(a + b)(a + b)(a + b) makes the pattern clearest because both terms are variables: a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴, with coefficients 1, 4, 6, 4, 1. The coefficient of akbn−k counts how many of the n brackets contributed the b — which is exactly what C(n, k) counts, and why the Binomial Coefficient Calculator produces the same numbers.
What collapses when you collect, and what does not
The 2ⁿ products are before collecting. How many survive depends on how much the brackets share.
(x + 1)(x + 2)(x + 3): 4 of the 8 products merged into others when like terms were collected, leaving 4 terms.
(x + 5)(x − 5) is the extreme case: 4 products collapse to 2, because the two middle ones are opposites and cancel outright. That is the difference of squares, and the cancellation is the whole of it.
The degree, by contrast, never collapses: it is always the sum of the bracket degrees, 3 here. That makes it a second, independent check — a leading term cannot cancel against anything, so a short degree means a real error rather than a collection.
Where this page stops, and what takes over
Every bracket must have exactly two terms. That is what a binomial is, and it is what makes 2ⁿ the count — a three-term bracket would contribute one of three choices, and the arithmetic would be 3 × 2 × 2 rather than a power of two.
So a bracket with any other number of terms is refused by name rather than quietly answered. The Multiplying Polynomials Calculator handles the general case, where the product count is the product of the term counts and the grid layout replaces the counting shortcut.
And for exactly two brackets with the four products named individually — First, Outer, Inner, Last — the FOIL Calculator is the page that does that, and it refuses three brackets for the same reason this one refuses three terms.
Sources and methodology
The distributive law, Vieta’s relations and the binomial theorem are standard; the references below carry the canonical statements.
Method. Every figure on this page comes from src/lib/multiplying-binomials.ts over the exact-rational term algebra in src/lib/algebra/parse.ts. The symmetric functions are computed by enumerating every k-subset of the roots rather than by reading coefficients back off the expansion, so the two columns are independent and their agreement is a real check. That engine is verified on every change against 41 hand-written assertions, including that the expansion always matches the general polynomial multiplier, that the degree equals the sum of the bracket degrees, and that the symmetric functions reproduce the coefficients. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
FOILMultiply two binomials with all four products named — First, Outer, Inner, Last — shown as a grid too, with difference of squares and perfect squares recognised.
Multiplying PolynomialsMultiply polynomials of any size in any number of variables, by grid, vertical or term-by-term layout, with the product count that catches a dropped term.
Binomial Coefficientn choose k exactly on big integers, by the multiplicative formula that divides as it goes and never builds a factorial — with Pascal's rule and permutations beside it.
Square of a BinomialExpand (a plus or minus b) squared with the middle term named and a numeric example of exactly what the (a+b)^2 = a^2+b^2 mistake costs, plus the area model that makes it stick.
Cubic EquationSolve any cubic exactly when it has a rational root — deflate and finish with the quadratic formula — and by the trigonometric form when it does not, with the discriminant saying which case you are in.
Rational ZerosList every rational root candidate from the rational root theorem, test each one, divide out the hits by synthetic division, and use the bound rules to strike the rest off without testing them.
Multiplying binomials is one of four polynomial operations and the one with the cleanest counting argument. Adding, Multiplying and Dividing Polynomials — the Four Operations covers the other three, including the product count that catches a dropped term in the general case.
Educational use disclaimer
This calculator multiplies between two and six binomials — brackets with exactly two terms each — exactly, in any number of variables. A bracket with any other number of terms is refused by name and pointed at the Multiplying Polynomials Calculator, because the 2ⁿ product count that this page is built on does not apply there.
Published the multiplying binomials page: two to six brackets, with the 2^n product count shown before the answer because that count is the check.
Adds the reading that separates it from both FOIL and the general polynomial multiplier — when every bracket is (x − r), the coefficients are the elementary symmetric functions of the roots, so the page turns roots into the polynomial that has them.
Identical brackets are recognised as a power and the coefficients matched against the corresponding row of Pascal's triangle.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.