Multiplying Polynomials Calculator

Count the products before you write any of them

There is one rule and it is short: every term in the first bracket multiplies every term in the second. That gives a number you can work out in advance — m terms times n terms is m×n products — and writing it down first is the single most useful habit in this operation.

• (2x + 3)(x − 5) is two by two: 4 products, which is the case FOIL names.
• (x + 2)(x² + 3x + 4) is two by three: 6 products, and FOIL has no name for the last two.
• (3x² − 2x + 1)(2x − 5) is three by two: 6 products, collecting to 4.

The reason to know the number in advance is that nothing later in the working reveals a missing product. If you write eight where there should be nine, the answer is a perfectly well-formed polynomial that is simply wrong, and no check on it will complain. Counting is the check.

What the count does not predict is how many terms survive. 6 products collect to 4 terms because several share a degree; 6 products in (2x + 3y)(x² − xy + y²) collect to 4. The count is a check on the products, not a prediction of the answer’s length.

The grid: the same rule, laid out so nothing can be skipped

Put the first factor’s terms down the side and the second factor’s across the top, and fill every cell with the product of its row and column. (x + 2)(x² + 3x + 4) makes a 2×3 table, and the answer is every cell added together.

It is not an alternative method. It is the distributive law with the bookkeeping made visual, and its advantage is entirely structural: an empty cell is obviously an empty cell, while a missing product in a line of algebra is invisible. On a 3×3, where there are nine products and no mnemonic, that difference is the whole game.

The grid also makes the collecting step easier to see. Terms of the same degree sit on the anti-diagonals of the table — the cells running from bottom-left to top-right — so gathering them is reading diagonally rather than hunting through a line.

The vertical layout, which is long multiplication

Stack the two polynomials the way you would stack two numbers, multiply the top by each term of the bottom in turn, and add the partial products down the column. For (3x² − 2x + 1)(2x − 5) that is 2 partial products, one per term of the second factor, aligned so like terms fall into the same column.

This is not an analogy to long multiplication with digits — it is the same algorithm. A number written in digits is a polynomial in 10, so 132 × 25 and (x² + 3x + 2)(2x + 5) are the same computation, and the only reason the digit version needs carrying is that the digits must stay below ten. Polynomials have no such constraint, which makes the algebraic version simpler than the arithmetic one rather than harder.

It suits paper better than the grid when the factors are long, because the alignment does the collecting for you: like terms are already in a column by the time you add.

Three factors: do them in pairs

(x + 1)(x + 2)(x + 3) can be expanded in one heroic pass with eight products, or in two easy passes. The second is better, and not only because it is less to hold in your head: multiplication is associative, so multiply any two of the factors, then multiply the result by the third, and the answer is the same however you group them.

(x + 1)(x + 2)(x + 3) multiplies to x³ + 6x² + 11x + 6. The calculator shows each partial product, because that intermediate is where a term goes missing — and if the final answer is wrong, the intermediate tells you which pass to re-check.

Choosing which pair to do first is worth a moment. Two factors that produce a short intermediate make the second pass smaller: if one factor is a monomial like the 5x in 5x(2x² − 3x + 1), doing that one last means distributing over fewer terms.

Two variables, and what “like terms” really means

Nothing changes with a second variable except what counts as like. (2x + 3y)(x² − xy + y²) gives 2x³ + x²y − xy² + 3y³: 6 products collecting to 4 terms, because only terms with identical variable parts combine.

That is the trap. x²y and xy² both have total degree 3, both contain x and y, and they are not like terms — the exponents must match variable by variable. Adding them is the multivariate equivalent of adding x² to x, and it is much easier to do by accident because the two look so similar on the page.

The ordering convention helps: write terms by descending power of the first variable, so x³, then x²y, then xy², then y³. Anything that belongs together is then adjacent, and anything adjacent that does not combine is visibly different.

The degree check, which takes five seconds

When polynomials multiply, their degrees add. A cubic times a quadratic is a quintic, always — the two leading terms multiply and nothing else can reach that degree, so nothing can cancel it. (3x² − 2x + 1)(2x − 5) has factors of degree 2 and 1, and the answer has degree 3.

So if your answer’s degree is not the sum of the factors’ degrees, a leading term was dropped or mis-multiplied, and you know exactly where to look. It is the cheapest check available and it catches the errors that matter most: a wrong leading term changes every downstream conclusion, from the shape of the graph to whether a factorisation is possible.

A second check for confidence rather than proof: substitute a number. Put x = 2 into the original product and into your answer; an identity holds for every x, so they must agree. It takes ten seconds and a disagreement is a definite error.

Sources and methodology

The distributive law and the degree rule are definitions and their consequences, written out above rather than cited — a reference would add nothing a reader could check that the argument does not. What is cited is the standard reference statement of polynomial algebra.

• NIST Digital Library of Mathematical Functions — §1.2, Elementary Algebra
• How we calculate — methodology

Method. Every figure on this page comes from src/lib/multiplying-polynomials.ts over the exact-rational term algebra in src/lib/algebra/parse.ts, where a term is a coefficient and a map of variable to exponent — so several variables are the normal case rather than an extension. Factors are multiplied left to right with each partial product kept, and the step-by-step answer is compared against the direct product of all the factors before anything is displayed. That engine is verified on every change against 77 hand-written assertions, including that the answer’s degree always equals the sum of the factors’ degrees and that the grid’s cells always re-collect to the printed answer. The count and the per-case breakdown are published on the formula verification page.

Read the guide

Multiplication is one of four operations and the others fail differently. Adding, Multiplying and Dividing Polynomials — the Four Operations puts all four together: the minus that reaches only the first term when subtracting, the product count that catches a dropped term here, the missing power that shifts a column when dividing, and the two checks — degree, and substituting a number — that work on every one of them.

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