HesapHanem

Polynomial Multiplication Calculator

Enter the coefficients of each polynomial, highest power first, separated by commas.

How it is calculated

Each polynomial is typed as a list of coefficients, highest power first, separated by commas or spaces. For a = [a0, …, am] and b = [b0, …, bn] the product has m + n + 1 coefficients, and ck = Σ ai × bj over all pairs with i + j = k (a convolution). The degree of the product is the sum of the two degrees. The result is printed as terms joined with +, so a negative coefficient appears as + -5x. Terms with a zero coefficient are skipped in the expression, but are kept in the coefficient list. Non-numeric entries make the tool return nothing.

Worked example

Multiply (2x + 3) by (x − 4), entered as 2,3 and 1,-4. Pair every coefficient: 2×1 = 2 for x², 2×(−4) + 3×1 = −8 + 3 = −5 for x, and 3×(−4) = −12 for the constant. The tool shows 2x^2 + -5x + -12, with coefficient list 2, -5, -12 and degree 2.

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Common mistakes

  • Listing coefficients lowest power first. The tool reads the first number as the highest power, so 3,2 means 3x + 2, not 2x + 3.
  • Leaving out zero coefficients for missing powers. For x² + 1 you must enter 1,0,1; entering 1,1 is the polynomial x + 1.
  • Typing the expression itself, such as 2x+3, instead of coefficients. Only numbers are accepted, so letters make the result empty.
  • Reading + -5x as an error. It is just the output style for a negative coefficient and means −5x.

About this calculator

Multiplying two polynomials means multiplying every term of the first by every term of the second and then combining like terms — the same distributive process taught as FOIL for two binomials, generalized to any degree. Given coefficient lists ordered from the highest power down, the result's coefficients come from convolving the two input lists: the coefficient of each resulting power is the sum of every pair of input coefficients whose exponents add up to that power.

This calculator takes the coefficients of two polynomials, highest degree first, and returns the fully expanded product polynomial, saving the error-prone manual bookkeeping of larger cross-multiplications by hand. Algebra and precalculus students checking homework, teachers generating answer keys, and engineers doing polynomial-based signal or control-system work — where multiplying polynomials is mathematically the same operation as convolving two signals — use it to verify expansions quickly.

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Comments (1)

  • Editorial team

    You can check any product without redoing the algebra: add the coefficients of each input, multiply those sums, and compare with the sum of the output coefficients. For 2,3 times 1,-4 that is 5 times -3 = -15, and 2 + -5 + -12 is also -15. A mismatch means a typo in an input. This catches slips but not offsetting ones. Do you want a division or evaluation tool alongside this one?

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