Binomial Expansion Calculator

Expand every term. See the pattern.

From the general rule to every individual term.

Exponent—

Binomial terms—

Pascal row sum—

Exact symbolic expansion

Finite binomial theorem

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Numeric check · x = 1, y = 1—

Pascal row · n = 4—

Step-by-step solution

Enter valid terms to see how the result is found.

Pascal’s triangle

Row n = 4 highlighted

Select a coefficient to explore its term. Other rows change n.

Text summary · PDF in member tools

Every term, explained

k	Binomial coefficient	Expanded term	At x = y = 1

The rule behind the result

{(a + b)}^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) a^{n - k} b^{k}

C(n,k) = n! / [k!(n − k)!]

Powers of the first term fall. Powers of the second rise. Pascal’s row supplies the coefficients; the original n + 1 terms may simplify further.

This tool handles finite non-negative integer powers. It does not compute an infinite binomial series.

Frequently asked questions

Where do the coefficients come from?

The coefficients C(n,k) count selections of k second terms from n factors. They form row n of Pascal’s triangle, counting the top 1 as row zero. Each interior entry is the sum of the two entries above it.

Can I expand a difference?

Yes. Enter a negative second term, such as -3y, to expand (2x − 3y)^n. The sign of each term follows the power of the negative input; it is not applied to the Pascal coefficients themselves.

What happens when n = 0?

The finite polynomial expansion uses the algebraic convention (a+b)^0 = 1: Pascal row zero contains one coefficient, 1. For a numeric base of zero, 0^0 is convention-dependent; this tool explicitly uses the polynomial convention 1 rather than a limiting interpretation.

Can I use variables and numbers?

Symbolic mode accepts one number or monomial per field: x, 2x, -3y, x^2, 0.5x or 1/2x. Products such as xy are supported, with up to four distinct letters across both terms. Numeric mode accepts numbers only. Parenthesized sums, functions, division by variables and negative variable powers are not supported.

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What limits apply?

The outer exponent n must be an integer from 0 to 30. Each input variable may have a whole-number power from 0 to 12. Decimal input supports up to six places; each numeric part may contain at most 12 digits. Large exact results can be scrolled. Larger Pascal triangles show a compact row window with the current row included.

Why can the final polynomial have fewer than n + 1 terms?

The binomial theorem generates n + 1 terms before simplification. Zero terms disappear and like powers combine. For example, (x + 2x)^4 simplifies to 81x^4 even though the binomial breakdown has five rows. The displayed binomial term count refers to those original rows.

How is the numeric check calculated?

Substitute a value for every input variable. The calculator evaluates each expanded term and sums them using exact rational arithmetic, then checks the same value by raising the sum of the two substituted input terms to n. Blank or invalid substitutions leave the symbolic expansion available.

Does this calculate Newton’s generalized binomial series?

No. This tool expands finite polynomials for non-negative integer exponents. Negative or fractional outer exponents generally require an infinite series and convergence conditions, which are outside this calculator.

What does the Binomial Expansion calculate?

Expand every term. See the pattern.

Which values should I enter?

What formula does this calculator use?

(a+b)^n = Σ C(n,k)a^(n−k)b^k for k=0…n, with C(n,k)=n!/[k!(n−k)!]. The finite tool supports integer n from 0 to 30.

Why might my result differ from another source?

Check signs, input powers and whether like terms are combined. Exact fractions can look different from rounded decimals. Distinguish a symbolic polynomial from its value at a chosen substitution.

Does the result update automatically?

Yes. Inputs update the expansion, optional numeric check, Pascal row, table and expandable step-by-step solution together. Invalid expressions clear previous results.