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