Question

Write a program that can store a polynomial such as $$ p(x)=5 x^{10}+9 x^{7}-x-10 $$ as a list of terms. A term contains the coefficient and the power of \(x\). For example, you would store \(p(x)\) as $$ (5,10),(9,7),(-1,1),(-10,0) $$

Short answer

List representation: [(5, 10), (9, 7), (-1, 1), (-10, 0)]

Step-by-step solution

Identify Polynomial Components

First, we recognize that a polynomial is composed of terms, where each term has its own coefficient and exponent. The given polynomial is \(p(x) = 5x^{10} + 9x^7 - x - 10\). We need to express each term separately: \(5x^{10}, 9x^7, -x, -10\).

Extract Coefficients and Exponents

Next, for each term, separate the coefficient and power of \(x\). For \(5x^{10}\), the coefficient is 5 and the power is 10. For \(9x^7\), the coefficient is 9 and the power is 7. For \(-x\), the coefficient is -1 (since \(-x = -1 \times x^1\)) and the power is 1. For \(-10\), which is \(-10x^0\), the coefficient is -10 and the power is 0.

Store Terms as Tuples

Once coefficients and exponents are identified, store each term as a tuple (coefficient, power). The polynomial terms are now: \((5, 10), (9, 7), (-1, 1), (-10, 0)\). These tuples will represent the polynomial in a list.

Create List

Compile the tuples from the previous step into a single list. This list will be our desired representation of the polynomial. The list format will be \([(5, 10), (9, 7), (-1, 1), (-10, 0)]\).

Key concepts

Coefficient and Exponent

In any polynomial, you will encounter two key components: coefficients and exponents. These are crucial in determining the behavior and shape of the polynomial.
Let's break these down: the **coefficient** is the numerical part of a term. It tells you how many times that part of the polynomial is used. For instance, in the term \(5x^{10}\), 5 is the coefficient.
The **exponent** indicates the power to which the variable is raised. In \(5x^{10}\), the exponent is 10, showing the variable \(x\) is multiplied by itself 10 times. Understanding these parts helps in creating, evaluating, and manipulating polynomials effectively.

Tuple Representation

Tuples are a fundamental concept in Python programming when it comes to storing fixed data sets. They are similar to lists but with one key difference—they are immutable, meaning once a tuple is created, it cannot be changed.
In representing a polynomial, each term can be stored as a tuple containing two elements: the coefficient and the exponent. For example, the term \(5x^{10}\) is stored as the tuple \((5, 10)\).
Using tuples allows for a clear and structured approach to organizing polynomial terms. They maintain the order and integrity of data, making it easy to read and access specified values efficiently.

List Data Structure

In Python, a list is a versatile data structure that can host a collection of items, such as tuples. Lists are mutable, which means you can modify them—add, remove, or change elements—after their creation.
To represent a polynomial, a list can hold several tuples, each representing a term with its coefficient and exponent. For instance, the polynomial \(p(x) = 5x^{10} + 9x^7 - x - 10\) can be stored as the list: \([(5, 10), (9, 7), (-1, 1), (-10, 0)]\).
Lists provide a great way to manage and manipulate collections of data. You can easily iterate over them, making them perfect for calculations or modifications needed in polynomial computations.

Python Programming

Python is a powerful and user-friendly programming language that provides various ways to manage and represent complex data, such as polynomials. It supports diverse data structures, like lists and tuples, which are perfect for storing and working with polynomial data.
When programming in Python, one can create dynamic and efficient polynomial representations by leveraging these data structures. Python's simple syntax makes it accessible for beginners, while its extensive libraries offer advanced tools for seasoned programmers who may need to perform complex polynomial computations or manipulations.
Whether creating a basic polynomial evaluator or a more complex manipulator, Python provides the necessary functions and libraries to make the tasks smoother and more approachable.

Polynomial Terms

Polynomial terms are the building blocks of any polynomial expression. Each term in a polynomial is composed of a variable raised to an exponent and multiplied by a coefficient.
Understanding polynomial terms is essential in polynomial algebra and calculus, as it involves operations such as adding, multiplying, and differentiating polynomials. For example, in the expression \(p(x) = 5x^{10} + 9x^7 - x - 10\), each grouping, such as \(5x^{10}\), represents a term.
Recognizing and working with these terms correctly allows for effective manipulation and understanding of polynomial functions. By using concepts like collecting coefficients and aligning terms according to their exponents, one can streamline polynomial operations efficiently.