# Polynomials: Everything You Need To Know

1. A-level maths topics
2. Algebra
3. Polynomials

A polynomial is a mathematical expression consisting of constants and variables, along with addition, subtraction, multiplication and division operations. Polynomials are a type of mathematical expression that can help us explore, understand, and solve complex problems in many areas of mathematics and science. From physics to engineering, polynomials are used to model and predict the behavior of systems and phenomena. In this article, we will take a look at what polynomials are, how they are used, and why they are so important for students studying mathematics with the help of experienced tutors from https://www.theprofs.co.uk/maths-tutors/.

There are several different types of polynomials that can be used to describe different kinds of phenomena.

#### Linear polynomials

, for example, consist of one variable raised to the power of one, and all terms in the polynomial have a degree of one. Quadratic polynomials consist of one variable raised to the power of two and all terms have a degree of two or less.

#### Cubic polynomials

consist of one variable raised to the power of three, and all terms have a degree of three or less.

Higher-order polynomials contain terms with degrees greater than three. Factoring is an important tool in working with polynomials. Factoring involves breaking down a polynomial into its component parts, or factors. By factoring a polynomial, it can be used to solve equations and other mathematical problems.

For example, factoring a quadratic equation will allow you to find its roots, or solutions. Polynomials are used in a variety of real-world applications. In physics, they are used to describe the motion of objects, such as planets or falling objects. In engineering, they are used to describe electrical circuits and other complex systems.

In economics, they are used to model market behavior and predict future trends. They are also used in other fields such as chemistry, meteorology, and even art and music. Graphing is another important tool in working with polynomials. By graphing a polynomial, you can visualize how it behaves in different situations.

Different types of polynomials can be used to model different phenomena. For example, linear polynomials can be used to describe straight lines, while higher-order polynomials can be used to model more complex phenomena such as curved shapes or dynamic systems.

## Factoring Polynomials

Factoring polynomials is an important skill in algebra and mathematics. It is used to simplify equations and find their solutions.

Factoring involves breaking down a polynomial into its component parts, namely its coefficients and variables. By factoring a polynomial, you can solve equations, identify the roots of the equation, and understand the properties of the polynomial. For example, consider the equation x2 - 2x + 1 = 0. By factoring this equation, we can identify that the roots of this equation are x = 1 and x = 1.Thus, we can conclude that the equation has two real roots. Additionally, we can find out that this equation is a quadratic equation. Factoring polynomials can also help us understand the behavior of a polynomial in certain situations.

For example, if we factor a polynomial, we can determine whether it has any complex roots or not. We can also identify if a polynomial has any zeros or not. Factoring is also useful in solving equations with multiple variables. By factoring a polynomial, we can identify which variables are related to each other and how they interact.

This can help us solve equations with multiple variables more quickly and accurately.

## Real-World Applications

Polynomials play an important role in many real-world applications, from physics and engineering to economics and other fields. In physics, polynomials are used to describe the motion of objects, such as planets or particles. For example, Newton's law of gravity can be written as a polynomial equation, which describes the force of gravity between two objects. In engineering, polynomials are used to model electrical circuits and other systems.

For example, the mathematical model of an electrical circuit can be written as a polynomial equation that describes how the current flows through the circuit. In economics, polynomials are used to model the behavior of prices and other economic variables. For example, the Demand Curve, which describes the relationship between price and quantity demanded, can be written as a polynomial equation. Polynomials are also used in other fields, such as statistics and machine learning. In statistics, polynomials are used to describe the relationship between variables.

For example, a regression analysis can be used to determine a polynomial equation that describes how two variables are related. In machine learning, polynomials are used to create models that can make predictions based on data. For example, a polynomial equation can be used to create a model that can predict the stock market prices.

## Defining Polynomials

A polynomial is a mathematical expression made up of variables and coefficients. It is used to model a wide range of physical and mathematical phenomena, such as the motion of planets or the behavior of electrical circuits.

Polynomials are constructed by adding and subtracting terms, which are composed of constants and variables. Each variable is raised to a power, or exponent, which is a whole number. The total sum of the terms in a polynomial is called its degree. There are several types of polynomials, including linear, quadratic, cubic, and higher-order polynomials.

Linear polynomials are those with one variable and one degree, such as ax + b, where a and b are constants. Quadratic polynomials have two variables and two degrees, such as ax2 + bx + c, where a, b, and c are constants. Cubic polynomials have three variables and three degrees, such as ax3 + bx2 + cx + d, where a, b, c, and d are constants. Higher-order polynomials have four or more variables and four or more degrees.

Polynomials can be used to model many different physical and mathematical phenomena. For example, they can be used to describe the motion of planets or the behavior of electrical circuits. They can also be used to solve equations, calculate derivatives, and approximate functions.

## Graphing Polynomials

Graphing polynomials is an important tool for visualizing and understanding the behavior of polynomials.

By graphing polynomials, we can better understand their shape and how they behave in different scenarios. This knowledge can then be used to model different phenomena. There are several methods for graphing polynomials, including plotting points, finding zeros and using technology such as graphing calculators or computer software. When plotting points on a graph, the x-coordinates of the points are values that satisfy the polynomial equation and the y-coordinates are the corresponding values of the polynomial at those x-coordinates.

This method is useful for understanding the behavior of polynomials, but is often difficult to use for more complex equations. Finding zeros of polynomials is another popular technique for graphing them. A zero of a polynomial is a value of x which makes the polynomial equal to zero. By finding these zeros, we can identify intervals where the polynomial is either positive or negative.

We can then plot points on the graph and connect them to form the graph of the polynomial. Finally, modern technology such as graphing calculators or computer software can be used to graph polynomials quickly and accurately. These tools can be used to create visual representations of even very complex polynomials, allowing us to easily understand their behavior. Graphing polynomials is an essential tool for understanding and modeling various phenomena.

By understanding how different types of polynomials behave, we can create models that accurately predict different physical and mathematical phenomena. In this article, we explored the fundamentals of polynomials and discussed how they are used in various applications. We defined polynomials and discussed how to factor them, explored their real-world applications, and showed how to graph them. Understanding polynomials is important for many applications such as physics, engineering, computer science, and economics. By studying polynomials, we gain the ability to model and predict a wide range of physical and mathematical phenomena.

We hope that this article has encouraged readers to explore more topics related to polynomials.

##### Shahid Lakha

Shahid Lakha is a seasoned educational consultant with a rich history in the independent education sector and EdTech. With a solid background in Physics, Shahid has cultivated a career that spans tutoring, consulting, and entrepreneurship. As an Educational Consultant at Spires Online Tutoring since October 2016, he has been instrumental in fostering educational excellence in the online tutoring space. Shahid is also the founder and director of Specialist Science Tutors, a tutoring agency based in West London, where he has successfully managed various facets of the business, including marketing, web design, and client relationships. His dedication to education is further evidenced by his role as a self-employed tutor, where he has been teaching Maths, Physics, and Engineering to students up to university level since September 2011. Shahid holds a Master of Science in Photon Science from the University of Manchester and a Bachelor of Science in Physics from the University of Bath.