Question

Write a Scheme function to evaluate the polynomial $2x^3-5x+1$. Include the code for all the subfunctions you use.

Short answer

Define `evaluate-polynomial` in Scheme to compute the polynomial's value.

Step-by-step solution

Understand the Problem

We need to write a Scheme function that evaluates a given polynomial $2x^3-5x+1$ for a value of $x$. Essentially, we will create a function that computes the polynomial's value when inputting a specific number for $x$.

Define the Main Function

Define the main function `evaluate-polynomial`. This function will take one argument, $x$, and will calculate the value of the polynomial $2x^3-5x+1$ by using multiplication and addition.

Scheme Code Structure

Write the Scheme code for the `evaluate-polynomial` function following the structure: ```scheme (define (evaluate-polynomial x) (+ (+ (* 2 (* x x x)) (* -5 x)) 1)) )```

Break Down Calculations

In the Scheme code, the polynomial is broken into parts:- $2x^3$ is computed using `(* 2 (* x x x))`.- $-5x$ is computed using `(* -5 x)`.- The sum of these two results is then added to 1.

Explain Scheme Syntax

In Scheme, `define` is used to create functions. `(* a b)` multiplies $a$ and $b$, `(+)` operator is used for addition, while each operation follows prefix notation (operator comes before the operands).

Key concepts

Polynomial Evaluation

Polynomial evaluation is a method of calculating the value of a polynomial expression for a given value of its variable. In this exercise, the polynomial we are dealing with is:

• $2x^3-5x+1$

To evaluate this polynomial for a specific value of $x$, you need to substitute the value of $x$ into the polynomial equation. Once the substitution is made, perform the mathematical operations to find the result.
Here's the breakdown of how you evaluate the polynomial step-by-step:

• First, multiply $x$ by itself three times to get $x^3$.
• Multiply $x^3$ by 2.
• Then, multiply $x$ by -5.
• Finally, add the results from the above steps and add 1 to it.

This process allows you to convert the polynomial into a simple arithmetic expression that can be calculated easily.

Scheme Functions

Scheme functions are fundamental components of the Scheme programming language, which is known for its simplicity and elegance. In Scheme, a function is defined using the `define` keyword. This allows you to encapsulate a set of instructions that can be reused throughout your program. In the context of the polynomial evaluation task:

• The primary function is `evaluate-polynomial`; its role is to compute the value of the polynomial given a specific $x$.

Here's a simple example of a Scheme function:

• ```scheme (define (add-two-numbers a b) (+ a b)) ```

Scheme functions use prefix notation, where the operator precedes the operands. For example, to add two numbers, you would use the expression `(+ a b)` rather than `a + b`. This syntax structure is consistent across all operations, making the language very uniform and predictable in its behavior.
Functions in Scheme also use recursion extensively as a form of iteration, which adds to the expressive power of the language.

Programming Languages

Programming languages are formal languages comprising a set of instructions that produce various kinds of output. They enable developers to write programs that can perform specific tasks. Scheme is one of such languages, often preferred for academic purposes and research due to its minimalist design and focus on functional programming. Scheme is designed to have a minimalistic syntax, which makes it easier for beginners to learn. A few characteristics of Scheme include:

• It employs a functional programming approach, emphasizing the use of functions and immutability.
• Scheme uses prefix notation, which can simplify parsing due to its consistency.
• It facilitates recursion and has powerful abstraction capabilities.

With programming languages, there is a need to understand not just how to solve a problem, but also which language features best apply to solving specific types of problems.

Functional Programming

Functional programming is a paradigm wherein programs are built by composing and applying functions. It emphasizes expressions and declarations over statements. In the Scheme programming language, functional programming is used extensively. Here are some of its core principles:

• **Immutability**: Functional programming discourages changing states and mutable data. Data objects are immutable, meaning they cannot be modified after creation.
• **First-class and Higher-order functions**: Functions are treated as first-class citizens, meaning they can be passed as arguments, returned by other functions, and assigned to variables.
• **Recursion**: Since loops are largely replaced by recursion in functional programming, recursion is a primary control structure.

In the context of evaluating polynomials in Scheme, functional programming enables the creation of small, reusable functions. You can define sub-functions to handle different parts of the polynomial, enhancing readability and reusability. By focusing on small, defined operations and using functions to combine these operations, Scheme demonstrates how the functional programming paradigm is effectively implemented.