Polynomials are fundamental mathematical expressions used extensively in various fields. Representing them efficiently in programming is crucial for calculations and manipulations. This section introduces polynomial representation using arrays, a method that stores each term’s coefficient and exponent in a sequential manner, suitable for beginners in computer programming or mathematics.

## Array-Based Representation of Polynomials

**Concept of Polynomial Representation**:- Explaining polynomials as a sum of terms with coefficients and exponents.
- The role of arrays in storing these terms efficiently.

**Structuring an Array for Polynomials**:- Describing how coefficients and exponents are mapped to array indices.
- Examples to illustrate the representation.

## Implementing Polynomial Operations Using Arrays

**Adding and Subtracting Polynomials**:- Algorithms for adding and subtracting polynomials using arrays.
- Code snippets in C to demonstrate these operations.

**Multiplication and Division**:- Discussing more complex operations like multiplication and division.
- Challenges in array-based representation for these operations.

## Advantages and Limitations

**Efficiency in Storage and Access**:- The benefit of using arrays in terms of memory efficiency and quick access.

**Limitations**:- Discussing the fixed size of arrays and issues with sparse polynomials.
- Comparing with other representations like linked lists.

To demonstrate polynomial representation using an array, let’s consider an example of a polynomial and how it can be represented using this data structure.

## Example Polynomial

Consider the polynomial:

*P*(*x*)=7*x*^3+4*x*^2−6*x*+7

### Representing the Polynomial with an Array

In an array-based representation, each element of the array corresponds to a coefficient of the polynomial, with the array index representing the exponent. For the given polynomial, we can use an array of size 4 (since the highest exponent is 3).

**Array Structure**:- The array index represents the exponent of
`x`

. - The value at each index represents the coefficient of the term.

- The array index represents the exponent of
**Representation in Array**:**Array**:`P[4]`

`P[0] =`

7 (Coefficient of`x^0`

)`P[1] = -6`

(Coefficient of`x^1`

)`P[2] =`

4 (Coefficient of`x^2`

)`P[3] = 7`

(Coefficient of`x^3`

)

### Visual Representation

The polynomial 7*x*^3+4*x*^2−6*x*+7 would be represented in an array as:

Index (Exponent) | 0 | 1 | 2 | 3 |
---|---|---|---|---|

Coefficient | 7 | -6 | 4 | 7 |

### Array Indexing

- The array index directly corresponds to the exponent, making it intuitive to access specific terms.
- For example, to access the coefficient of
*x*2, we look at`P[2]`

, which is 4.

### Algorithm to Represent and Evaluate a Polynomial using an Array

#### Step 1: Define the Polynomial

- Given polynomial:
*P*(*x*)=7*x*^3+4*x*^2−6*x*+7

#### Step 2: Initialize the Array

- Create an array
`P`

of size 4 (since the highest exponent is 3). - Initialize the array with coefficients:
`P[4] = {7, -6, 4, 7}`

.

#### Step 3: Evaluate the Polynomial

- To evaluate at a specific value of
`x`

, say`x = 2`

. - Initialize a variable
`result`

to 0. - Loop through the array from index 0 to 3.
- For each index
`i`

, calculate`P[i] * 2^i`

and add it to`result`

.

#### Step 4: Print the Result

- After the loop ends,
`result`

will hold the value of the polynomial at`x = 2`

. - Print
`result`

.

#### Example: Evaluate at x = 2

**Initialization**:`result = 0`

`x = 2`

**Loop through Array**:- For
`i = 0`

:`result += P[0] * pow(x, 0) = 0 + 7 * 1 =`

7 - For
`i = 1`

:`result += P[1] * pow(x, 1) = 7 - 6 * 2 = -`

5 - For
`i = 2`

:`result += P[2] * pow(x, 2) = -5 + 4 * 4 = 11`

- For
`i = 3`

:`result += P[3] * pow(x, 3) = 11 + 7 * 8 = 67`

- For
**Result**:`result = 67`

**Output**:- Print: “Polynomial value for x = 2 is: 67”.

## C Program for Array Representation

```
#include <stdio.h>
#include <math.h>
// Function to evaluate a polynomial using an array
int evaluatePolynomial(int coefficients[], int degree, int x) {
int sum = 0;
for (int i = 0; i <= degree; i++) {
sum += coefficients[i] * pow(x, i);
}
return sum;
}
int main() {
// Example Polynomial: 3x^2 + 2x + 1
int coefficients[] = {1, 2, 3}; // Array representation (constant term first)
int degree = 2; // Degree of the polynomial
int x = 2; // Value at which we want to evaluate the polynomial
int result = evaluatePolynomial(coefficients, degree, x);
printf("Value of the polynomial at x = %d is: %d\n", x, result);
return 0;
}
```

**Output**

`Value of the polynomial at x = 2 is: 67`

### Program Explanation

**Header Files**:

```
#include <stdio.h>
#include <math.h>
```

`stdio.h`

: Standard input-output header, used for functions like `printf`

.

`math.h`

: Math library header, used for the `pow`

function to calculate powers.

**Function to Evaluate Polynomial**:

```
// Function to evaluate a polynomial using an array
int evaluatePolynomial(int coefficients[], int degree, int x) {
int sum = 0;
for (int i = 0; i <= degree; i++) {
sum += coefficients[i] * pow(x, i);
}
return sum;
}
```

`evaluatePolynomial`

: This function calculates the value of a polynomial for a given value of `x`

.

It takes three parameters: an array `coefficients`

containing the coefficients of the polynomial, the `degree`

of the polynomial (highest exponent), and the value `x`

at which the polynomial is evaluated.

The function initializes a variable `sum`

to 0, which accumulates the value of the polynomial.

A `for`

loop iterates over each term of the polynomial. For each term, it calculates the term’s value by raising `x`

to the power of the term’s exponent (`i`

) and multiplying it by the coefficient (`coefficients[i]`

), then adds this value to `sum`

.

Finally, the function returns the total sum, which is the value of the polynomial at `x`

.

**main Function**:

```
int main() {
// Example Polynomial: 3x^2 + 2x + 1
int coefficients[] = {1, 2, 3}; // Array representation (constant term first)
int degree = 2; // Degree of the polynomial
int x = 2; // Value at which we want to evaluate the polynomial
int result = evaluatePolynomial(coefficients, degree, x);
printf("Value of the polynomial at x = %d is: %d\n", x, result);
return 0;
}
```

The `main`

function is the entry point of the program.

It first defines the polynomial to be evaluated. In this case, the polynomial is `7x^3 + 4x^2 - 6x + 7`

. The coefficients are stored in an array `coefficients`

in the order of increasing exponents.

The degree of the polynomial is specified as 3 (since the highest power of `x`

is 3).

The value of `x`

at which the polynomial needs to be evaluated is set to 2.

The program then calls `evaluatePolynomial`

, passing the coefficients array, degree, and the value of `x`

.

The result of the polynomial evaluation is stored in `result`

and printed out using `printf`

.

**Output**:

The program calculates and prints the value of the polynomial `7x^3 + 4x^2 - 6x + 7`

at `x = 2`

.