Evaluate polynomials with Horner’s rule from descending coefficients, calculate the derivative at x, or perform synthetic division to find the quotient and remainder.

Enter coefficients in descending powers, including 0 for missing powers.

1–16 numbers, separated by commas or spaces. Use a decimal point. Commas separate coefficients, never digit groups.

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Horner’s Rule Formula

The following formula is used to calculate the value of a polynomial using Horner’s Rule.

P(x) = a0 + a1 × x + a2 × x² + a3 × x³ + ... + an × xⁿ

Variables:

  • P(x) is the value of the polynomial at x
  • a0, a1, a2, …, an are the coefficients of the polynomial; x is the value at which the polynomial is evaluated
  • n is the degree of the polynomial

To calculate the value of a polynomial using Horner’s Rule, start with the highest-degree coefficient an (not an*x^n). For each subsequent term, multiply the result so far by x and add the coefficient of the next lower-degree term. Repeat this process until you reach the constant term (a0). The result is the value of the polynomial at x.

What is Horner’s Rule?

Horner’s Rule is a mathematical algorithm used for efficient computation of polynomial expressions. It simplifies the process of evaluating a polynomial at a given value by reducing the number of multiplications and additions required. The rule works by reorganizing the polynomial into nested multiplication, essentially transforming it into a sequence of simpler calculations. This method is named after British mathematician William George Horner.

How to Calculate Horner’s Rule?

The following steps outline how to use Horner’s Rule to evaluate a polynomial:

  1. Gather the coefficients in descending order (an, …, a1, a0), including0 for missing powers, and enter x. Commas or whitespace separate coefficients; commas are not digit grouping.
  2. Next, write down the formula: P(x) = a0 + a1*x + a2*x^2 + a3*x^3 + … + an*x^n.
  3. Start with an; multiply the running value by x and add the next coefficient, not the next full polynomial term.
  4. Repeat step 3 for each term, working your way down to the lowest-degree term (a0).
  5. Finally, the result of the last addition is the value of the polynomial at x (P(x)).

Example Problem:

Use the following variables as an example problem to test your knowledge:

Descending coefficients (an, …, a0) = 3, 2, 1, representing P(x) = 3x² + 2x + 1

Value (x) = 4; P(4) = (3*4 + 2)*4 + 1 = 57 and P′(4) = 6*4 + 2 = 26.