Horner's Method Calculator

How it is calculated

Coefficients are read from a comma-separated list, highest power first, and the polynomial is evaluated by nesting: start with r = 0, then for each coefficient c in order do r = r × x + c. After the last coefficient, r equals anxn + … + a1x + a0. The degree shown is the number of coefficients minus 1, not counting leading zero coefficients. The value and the echoed x are shown to 4 decimals. An empty slot between commas (1,,2) counts as a zero coefficient and one trailing comma is ignored. A non-numeric entry returns nothing, and a missing power must be written as 0 (for example 1,0,0,-1 for x³ − 1), because the position in the list sets the power. A result too large to represent also shows nothing.

Common mistakes

• Entering the coefficients lowest power first. The list is read highest-degree first, so -1,2,-6,2 describes a different polynomial from 2,-6,2,-1.
• Skipping a zero coefficient. For x⁴ − 1 you must enter 1,0,0,0,-1; entering 1,-1 gives x − 1 instead.
• Leaving an empty slot by accident. 1,,2 is read as 1,0,2 (x² + 2, degree 2), so a doubled comma silently inserts a zero coefficient. At x = 2 it gives 6, not the 4 you would get from 1,2.
• Separating coefficients with spaces or semicolons. Only commas split the list, so 2 -6 2 is rejected as not a number.
• Trusting the degree when the first coefficient is 0. Leading zero coefficients do not raise the degree: 0,0,3 is shown as degree 0 because it is just the constant 3, so check that the degree matches the polynomial you meant.