Enter the coefficients and the value at which the polynomial is evaluated into the calculator to determine the value of the polynomial using Horner’s Rule.

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*x^2 + a3*x^3 + ... + an*x^n

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 term (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 a 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. First, gather the coefficients of the polynomial (a0, a1, a2, …, an) and the value at which the polynomial is evaluated (x).
  2. Next, write down the formula: P(x) = a0 + a1*x + a2*x^2 + a3*x^3 + … + an*x^n.
  3. Starting from the highest degree term (an*x^n), multiply the coefficient (an) by the value (x) and add the result to the next lower degree 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:

Coefficients (a0, a1, a2, …, an) = 3, 2, 1

Value (x) = 4