What Are Polynomials?

Polynomials are algebraic expressions with fixed numbers called constants and terms that can change called variables, connected through operators like addition, multiplication, etc., and the power on the variables are always non-negative integers. Polynomials are almost used in every field that needs math. This blog will explain the basics of polynomials and some terms related to polynomials.

Structure of a polynomial

What are polynomials, and why do we need them

Let’s take the example of 2x4 + 3x2 + 7x + 4

  • 4 is a fixed number, hence constant.
  • The value of x can be any number; hence, the value is not constant. We call it a variable.
  • The numbers in superscript are the powers (number of times a variable is multiplied by itself) to which the variable x has been raised.
  • Terms are individual units of the equation, connected through operators like addition or multiplication.
  • The fixed number before the variable is known as the leading coefficient.
  • When nothing is written before the variables, we consider the leading coefficient as 1.
  • A polynomial can have two or more different variables.

Before going on to the next topic, let’s use what we have learned to identify polynomials

Which of the following is/are a polynomial:

  1. 9x3 + 5x-1 + x + 6
  2. 5x6 + 4x5 + xx + 7
  3. 7x5 + 6√x+ 8
  4. 10x + 5y

Solution: 1 and 4

We can cancel options 1 and 3 since terms 5x-1 is a negative integer, and 6√x has a non-integer exponent, respectively.  When a variable is raised to the power ½, it’s the square root of the variable. Options 1 and 2 are polynomials, as they have positive integers as the exponents of the variable.

Standard form of a polynomial

A standard form is the blueprint of a polynomial. You have the design, and now you need to build upon that.

The standard form of a polynomial is:

P(x) = anxn + an-1xn-1 +an-2xn-2 + . . .  + a1x + a0  

  • n in superscript represents the power on the variable, and n in subscript means that we are at the nth term where n is any positive integer.
  • The leading coefficients are denoted by ‘a’.
  • The variable we have considered is x.
  • P(x) is the function of the polynomial. A function is a mathematical vending machine. If you give a value to a function, it provides an output based on the equation.

Let’s try out an example problem. Write down the standard form for the polynomial:

4x2 + x3 + 7x + 5x4 + 6

Solution: According to the standard form, the polynomial should be arranged in a decreasing order of the powers of their degree. So, the terms will be placed as follows:

5x4+ x3 + 4x2 + 7x + 6

How to find the degree of a polynomial

The degree of a polynomial refers to the highest power of a variable in the equation. The term with the highest degree is placed first, followed by the next ones in a decreasing order of their degree.

Degree of an equation with one variable

The degree is the same as the highest power of the variable. For example, the degree of the polynomial 5x4+ x3 + 4x2 + 7x + 6 is 5, since the highest power on x is 5.

Degree of an equation with double variable

An equation with two variables poses a challenge in finding the degree. Let’s take a look at the polynomial x2y2 + 5x2y + 14x3+2. Here, we check the sum of the exponents of x and y.

Therefore,

x2y2= 2+2= 4

5x2y= 2+1= 3

14x3= 3

The maximum sum of the exponents is 4. Therefore, the degree of the polynomial is 4.

Types of polynomials

Polynomials are categorized based on the number of terms a polynomial has. If the polynomial has one term, we call it a monomial. If there are two terms, we call them binomials, and trinomials if there are three.

A polynomial might not even have a variable and only has a constant. These are known as constant polynomials.

We can also classify the terms into two categories: like and unlike.

Like terms have the same variable following the leading coefficient. Unlike terms, they have different variables or are the same variables raised to different powers.

Now you know what polynomials are. We can move on to actual calculations using the basics we have covered. The next blog will discuss polynomial calculations (addition, subtraction, multiplication, and division).

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