Polynomials – Definition With Examples

As we voyage on this mathematical journey together here at Brighterly, one fundamental concept that will recurrently surface in our discussions is polynomials. Whether you’re venturing into algebra for the first time or grappling with the intricate labyrinths of calculus, polynomials are ubiquitous. But what are they exactly, and how do they form the bedrock of so many mathematical concepts and real-life applications?

At their core, polynomials are mathematical expressions composed of variables (commonly represented as ‘x’), coefficients, and exponents that are whole numbers (0, 1, 2, 3, etc.). The Greek roots of the term – ‘poly’ (meaning ‘many’) and ‘nomial’ (meaning ‘terms’) – encapsulate this concept beautifully. Polynomials can indeed comprise a single term or an amalgamation of many. From the representation of trends in data analysis to the modeling of natural phenomena in the sciences, polynomials play a pivotal role. Today, let’s uncover the magic behind these essential mathematical tools and learn how to wield them with finesse.

What Are Polynomials?

Polynomials are fundamental objects in mathematics that crop up in a variety of subjects and real-world applications. These powerful tools can represent trends, model natural phenomena, or serve as a stepping stone to more complex mathematical concepts. But what exactly are polynomials?

At its core, a polynomial is a mathematical expression made up of variables (often represented by ‘x’), coefficients, and exponents that are whole numbers (0, 1, 2, 3, and so on). The term ‘polynomial’ comes from the Greek words ‘poly’ (which means ‘many’) and ‘nomial’ (which means ‘terms’), and it is a suitable name as polynomials can have one or many terms. Whether you’re working on simple algebraic problems or complex calculus, polynomials are bound to make an appearance.

Definition of Polynomials

A polynomial is formally defined as an expression of the form:

a_n*x^n + a_(n-1)*x^(n-1) + ... + a_2*x^2 + a_1*x + a_0 where a_n, a_(n-1), ..., a_2, a_1, a_0 are the coefficients, x is the variable, and n is the highest exponent, known as the degree of the polynomial. The coefficients are real numbers, and n must be a non-negative integer.

For example, 3x^2 - 4x + 7 is a polynomial. Here, 3, -4, and 7 are coefficients, x is the variable, and 2 is the degree of the polynomial.

Elements of a Polynomial

Each polynomial consists of certain elements, including terms, coefficients, variables, exponents, and degree. A term in a polynomial is a single part of the expression, separated by plus or minus signs. The coefficient is the numerical part of each term. The variable (commonly ‘x’) is the letter in the term. The exponent is the power to which the variable is raised. The degree of a polynomial is the highest power of the variable in that polynomial.

For example, in the polynomial 5x^3 - 4x^2 + 3x - 2, there are four terms, coefficients are 5, -4, 3, -2, the variable is x, exponents are 3, 2, 1, 0 respectively, and the degree is 3.

Properties of Polynomials

Polynomials possess several fascinating properties that make them crucial for solving complex mathematical problems. For example, polynomials are closed under addition, subtraction, and multiplication, which means if you add, subtract, or multiply two polynomials, you will get another polynomial. Polynomials can be factored, and the Zero-Product Property is often used to solve polynomial equations.

Types of Polynomials

There are various types of polynomials based on the number of terms and the degree of the polynomial. The most common ones include Monomials, Binomials, Trinomials, and Polynomials of higher degree. Each type has unique properties and behaves differently when graphed or factored.

Difference Between Different Types of Polynomials

The difference between different types of polynomials primarily lies in their structure, the number of terms they have, and their degree. Monomials have one term, Binomials have two, Trinomials have three, and so on. Additionally, the degree of the polynomial, or the highest power of the variable in the polynomial, can also differentiate these types.

Equations of Polynomials

An equation of a polynomial sets the polynomial equal to zero or another polynomial. Polynomial equations often appear in algebra and calculus problems, and there are many techniques, such as factoring, the Rational Root Theorem, and synthetic division, used to solve these equations.

Writing Equations of Polynomials

Writing equations of polynomials requires understanding of the polynomial’s roots or zeros, the degree of the polynomial, and the leading coefficient. With this information, one can write the polynomial equation using the factored form or the expanded form.

Polynomial Functions and Their Graphs

Polynomial functions are those functions that are defined by polynomials. The graphs of polynomial functions are smooth, continuous curves that can have hills, valleys, and many points where they cross the x-axis (known as zeros or roots). The shape of the graph is determined by the degree of the polynomial and the sign of the leading coefficient.

Practice Problems on Polynomials

To solidify your understanding of polynomials, here are a few practice problems:

  1. Write a polynomial that has a degree of 2, a leading coefficient of 1, and roots of 3 and -3.
  2. Identify the degree and leading coefficient of the polynomial 2x^4 – 3x^3 + x – 1.
  3. Factor the polynomial x^2 – 5x + 6.

Conclusion

As we round off our journey through the world of polynomials, it’s clear to see why these mathematical expressions are a cornerstone in numerous fields. Their flexibility and applicability make them a fundamental concept in mathematics, from algebraic equations to intricate calculus problems.

At Brighterly, we firmly believe in the power of understanding these foundational elements. A strong grasp of polynomials not only aids in solving complex mathematical problems but also enables the development of logical thinking and problem-solving skills that extend beyond the realms of mathematics.

So, as we bid farewell to polynomials for now, remember that this is only the tip of the mathematical iceberg. There are countless other exciting concepts waiting to be discovered. But with polynomials in your mathematical toolbox, you’re well on your way to mastering even the most formidable mathematical challenges that lie ahead. Here at Brighterly, we’re excited to guide you every step of the way!

Frequently Asked Questions on Polynomials

What is the degree of a polynomial?

The degree of a polynomial is essentially the highest power of the variable within the polynomial. For example, in the polynomial 3x^2 - 4x + 7, the degree is 2, because the highest power of the variable x is 2. The degree of a polynomial provides valuable information about the polynomial, including the number of solutions or roots it has and its general shape when graphed.

What are the roots or zeros of a polynomial?

Roots, also known as zeros, of a polynomial are the values of the variable that make the polynomial equal to zero. For example, in the polynomial equation x^2 - 4 = 0, the roots are 2 and -2, because substituting these values in place of x makes the entire equation equal to zero. The roots of a polynomial can be found using various methods such as factoring, the quadratic formula, or synthetic division, depending on the complexity of the polynomial.

How do you factor a polynomial?

Factoring a polynomial involves expressing the polynomial as a product of simpler polynomials. For instance, the polynomial x^2 - 5x + 6 can be factored as (x - 2)(x - 3). Factoring is an essential process in simplifying polynomials and solving polynomial equations. Various methods, such as factoring by grouping, factoring trinomials, and factoring by using special formulas, can be used to factor polynomials, depending on the structure of the polynomial.

Information Sources
  1. Wolfram MathWorld
  2. The National Center for Biotechnology Information
  3. The U.S. Department of Education

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