Expand Polynomials Calculator

Enter any polynomial expression into the Expand Polynomials Calculator and get the fully expanded result with a step-by-step breakdown. Type expressions like (2x + 3)(x − 5), 3(x + 6)², or (x + y)³ into the expression field, choose your expansion method (FOIL or Binomial Theorem), and the calculator returns the expanded polynomial, degree, number of terms, and leading coefficient — all shown clearly so you can follow the working.

Enter expressions using * for multiplication, ^ for powers. Example: (x+2)*(x-3) or (x+1)^3

Auto-detect chooses the best method based on your expression.

The variable used in your expression (usually x).

Results

Expanded Polynomial

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Degree of Polynomial

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Number of Terms

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Leading Coefficient

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Constant Term

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Method Used

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Coefficient of Each Term

Results Table

Frequently Asked Questions

What does it mean to expand a polynomial?

Expanding a polynomial means removing parentheses by multiplying out all factors so the expression is written as a sum of individual terms. For example, (x + 2)(x + 3) expands to x² + 5x + 6. It is essentially the reverse of factoring.

What is the FOIL method and when do I use it?

FOIL stands for First, Outer, Inner, Last — it's a systematic way to multiply two binomials. For (ax + b)(cx + d), you multiply: First (ac·x²), Outer (ad·x), Inner (bc·x), Last (bd), then combine like terms. FOIL only applies to exactly two binomials; for longer polynomials use the distributive property.

How do you expand using the Binomial Theorem?

The Binomial Theorem expands (a + b)ⁿ using the formula Σ C(n,k) · aⁿ⁻ᵏ · bᵏ for k from 0 to n, where C(n,k) are binomial coefficients (Pascal's Triangle). For example, (x + 1)³ = x³ + 3x² + 3x + 1. This method works for any positive integer power n.

How do I expand a polynomial with more than two factors?

Expand two factors at a time using FOIL or the distributive property, then multiply the result by the next factor. For example, (x+1)(x+2)(x+3): first expand (x+1)(x+2) = x²+3x+2, then multiply that by (x+3) to get x³+6x²+11x+6.

Why would you want to expand an expression?

Expanding makes it easier to add, subtract, compare, or differentiate polynomials. It converts a factored form into standard form, which is necessary for many algebraic operations like finding roots, simplifying equations, performing polynomial long division, and calculus differentiation.

What are common mistakes when expanding polynomials?

The most frequent errors include forgetting to distribute a negative sign (e.g. -(x+2) = -x-2, not -x+2), skipping the Outer and Inner terms in FOIL, and incorrectly applying exponent rules such as writing (x+2)² as x²+4 instead of x²+4x+4.

How do you expand polynomials with exponents?

For expressions like (x+a)ⁿ, use the Binomial Theorem or repeatedly apply the distributive property. For instance, (x+2)² = (x+2)(x+2) = x²+4x+4. For higher powers, Pascal's Triangle gives the binomial coefficients directly — the nth row gives the coefficients for (a+b)ⁿ.

What is the degree of the expanded polynomial?

The degree is the highest power of the variable in the expanded polynomial. When you multiply two polynomials, the degree of the result equals the sum of their individual degrees. For example, multiplying a degree-2 by a degree-3 polynomial gives a degree-5 result.

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