1. Introduction

Polynomial equations are a fundamental element in algebra and calculus, with numerous applications in science and engineering. In programming, computing the value of a polynomial equation for a given variable involves evaluating the terms with various powers based on the given coefficients. Python, with its robust arithmetic capabilities, can perform this task efficiently.

A polynomial equation is a mathematical expression consisting of variables, coefficients, and exponents. The general form of a polynomial equation is an*x^n + an-1*x^(n-1) + ... + a2*x^2 + a1*x + a0, where an, an-1, ..., a0 are the coefficients, x is variable, and n is the degree of the polynomial.

2. Program Steps

1. Define the coefficients of the polynomial equation.

2. Define the value of the variable x.

3. Evaluate the polynomial using the coefficients and the variable value.

4. Print the result of the evaluation.

3. Code Program

``````# Function to compute the value of a polynomial equation
def compute_polynomial(coefficients, x):
# Initialize result of the polynomial
result = 0
# Compute each term of the polynomial and add to the result
for power, coeff in enumerate(coefficients):
result += coeff * (x ** power)
return result

# Coefficients of the polynomial a0, a1, a2, ..., an from the lowest degree to the highest
coefficients = [1, -4, 3]  # Represents 1 - 4x + 3x^2
# Value of the variable x
x_value = 2
# Compute the polynomial
polynomial_result = compute_polynomial(coefficients, x_value)
# Print the result
print(f"The value of the polynomial for x = {x_value} is: {polynomial_result}")
``````

Output:

```The value of the polynomial for x = 2 is: 7
```

Explanation:

1. The compute_polynomial function takes a list of coefficients and a value for x.

2. result is initialized to 0 to accumulate the sum of each term in the polynomial.

3. The function iterates over coefficients using enumerate, which provides both the power (index) and the coeff (value).

4. For each term, the function calculates coeff (x * power) and adds it to result.

5. After all terms are evaluated, result holds the value of the polynomial.

6. coefficients are defined as [1, -4, 3], which corresponds to 1 - 4x + 3x^2.

7. x_value is set to 2, and the polynomial is computed by calling compute_polynomial.

8. The print statement displays the result, stating that the value of the polynomial for x = 2 is 7.