Hello can you please help me do this exercise. Can you do part 1, 2, 3, 4 and 5 (marked in red please) **Exercise 4:**The following algorithm computes the sum of a polynomial of power n. For an explanation of what a polynomial is, consult: [Polynomial Explanation](https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-polynomial-2009-1.pdf)The polynomial is of the generic form:\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0 \]We will assume that the coefficient values \( a_0 \) through \( a_n \) are all known and constant and stored in an array. This means that the evaluation of the polynomial has only the value \( x \) as its input and will return the resulting polynomial value as its output.The standard evaluation algorithm is straightforward:```Evaluate (x) {// x the value to use for evaluation of the polynomial result = a[0] + a[1] * x xPower = x for i = 2 to n do xPower = xPower * x result = result + a[i] * xPower end for return result.}```1. **Find the number of additions and multiplications outside the for loop.**2. **Find the number of additions inside the for loop.**3. **Find the number of multiplications inside the for loop.**4. **Find the number of times the for loop executes.**5. **Find the number of operations of the function Evaluate(x).** **Polynomial Evaluation Using Factorization**To efficiently evaluate a polynomial, it can be factored into the following form:\[ p(x) = ((\ldots((a_nx + a_{n-1})x + a_{n-2})x + \ldots + a_2)x + a_1)x + a_0 \]This can be expressed algorithmically as:**BetterEvaluate(x)**- **// x**: the value to use for evaluation of the polynomial- **result = a[n]**For **i = n-1** down to **0** do - **result = result * x** - **result = result + a[i]**End ForReturn **result**.**Exercises**1. Answer questions 1-5 regarding the algorithm BetterEvaluate(x).2. How many multiplications were saved by applying BetterEvaluate(x) rather than Evaluate(x)?**Extra Credit (20 pts)**- Implement each of the algorithms for **x = 2** and \(\text{a[10]} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).

Answered: The polynomial is of the generic form:…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Hello can you please help me do this exercise. Can you do part 1, 2, 3, 4 and 5 (marked in red please)

**Exercise 4:**

The following algorithm computes the sum of a polynomial of power n.
For an explanation of what a polynomial is, consult:
[Polynomial Explanation](https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-polynomial-2009-1.pdf)

The polynomial is of the generic form:

\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0 \]

We will assume that the coefficient values \( a_0 \) through \( a_n \) are all known and constant and stored in an array. This means that the evaluation of the polynomial has only the value \( x \) as its input and will return the resulting polynomial value as its output.

The standard evaluation algorithm is straightforward:

```
Evaluate (x) {
// x the value to use for evaluation of the polynomial
result = a[0] + a[1] * x
xPower = x
for i = 2 to n do
xPower = xPower * x
result = result + a[i] * xPower
end for
return result.
}
```

1. **Find the number of additions and multiplications outside the for loop.**
2. **Find the number of additions inside the for loop.**
3. **Find the number of multiplications inside the for loop.**
4. **Find the number of times the for loop executes.**
5. **Find the number of operations of the function Evaluate(x).**

**Polynomial Evaluation Using Factorization**

To efficiently evaluate a polynomial, it can be factored into the following form:

\[ p(x) = ((\ldots((a_nx + a_{n-1})x + a_{n-2})x + \ldots + a_2)x + a_1)x + a_0 \]

This can be expressed algorithmically as:

**BetterEvaluate(x)**

- **// x**: the value to use for evaluation of the polynomial
- **result = a[n]**

For **i = n-1** down to **0** do
- **result = result * x**
- **result = result + a[i]**

End For

Return **result**.

**Exercises**

1. Answer questions 1-5 regarding the algorithm BetterEvaluate(x).
2. How many multiplications were saved by applying BetterEvaluate(x) rather than Evaluate(x)?

**Extra Credit (20 pts)**

- Implement each of the algorithms for **x = 2** and \(\text{a[10]} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).

Expert Solution

Step 1: Introduce polynomial evoulution:

Polynomial evaluation is a fundamental mathematical operation where a polynomial expression is computed for a given value of its variable. The generic form of a polynomial consists of coefficients stored in an array, and a standard evaluation algorithm is used to calculate the polynomial's value efficiently. Understanding the number of additions, multiplications, and loop iterations in the evaluation process is essential for optimizing computational efficiency.

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