Polynomial – Definition, Types, Properties, Operations, Examples | How do you Identify Polynomials?

Polynomial is an algebraic expression that consists of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations like addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable. These are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math. For example, 2x + 9 and x2 + 3x + 11 are polynomials.

Look at this entire article in order to understand the 10th Grade Math polynomial concept in a better way. You will learn about what is polynomial, the definition of polynomial, the types of polynomial, the degree of a polynomial, properties of polynomial, terms, operations of polynomial, factorization of polynomial, Equations, functions, examples, and so on.

What is a Polynomial?

A polynomial is a type of expression. An expression is a mathematical statement that doesn’t have an equal-to sign (=).

Definition of Polynomial

A polynomial is a type of algebraic expression in which all variable’s exponents should be a whole number. The variables exponent in any polynomial has to be a non-negative integer. It will comprise constants and variables, but we cannot perform division operations by a variable in polynomials. Generally, the Polynomials are the sum or difference of variables and exponents. Each part of the polynomial is known as a “term”. An example of a Polynomial is 4x2+3x−7.

(1) Real polynomial:
f(x) = a0 + a1x + a2x2 + a3x3 + …….. + anxn is called real polynomial of real variable x with real coefficients.
Example: 3x3 – 4x2 + 5x – 4, x2 – 2x + 1 etc. are real polynomials.

(2) Complex polynomial:
f(x) = a0 + a1x + a2x2+ a3x3 + …….. + anxn is called complex polynomial of complex variable x with complex coefficients.
Example: 3x2 – (2+4i)x + (5i-4), x3 – 5ix2 + (1+2i)x + 4 etc. are complex polynomials.

Polynomial Examples

For understanding, consider an example for 3x2 + 5. In the given polynomial, certain terms are there we need to understand. Here, x is known as the variable. 3 which is multiplied by x2 has a special name. We denote it by the term “coefficient” and 5 is known as the constant. The power of the x variable is 2. Below given are a few expressions that are not examples of a polynomial, those are:

  • 2x-2: Here, the exponent of the variable is -2
  • 1/(y + 2): This is not an example of a polynomial since division operation in a polynomial cannot be performed by a variable.
  • √(2x): The exponent cannot be a fraction(here, 1/2) for a polynomial.

Standard Form of Polynomial

The standard form of a polynomial is nothing but to write a polynomial in the descending power of the variable. Let us express an example of the standard form of a polynomial is 5 + 2x + x2 in the standard form. To express the polynomial in standard form, we will first check the degree of the polynomial.

  • In the given polynomial, the degree is 2. First, we write the term containing the degree of the polynomial.
  • Then, we will check if there is a term with the exponent of variable less than 2, i.e., 1, and note it down next.
  • Now write the terms with the exponent of the variable as 0, which is the constant term. Therefore, the expression 5 + 2x + x2 is in standard form it can be written as x2 + 2x + 5.

Remember that in the standard form of a polynomial, the terms are always written in decreasing order of the power of the variable, here, x.

Terms of a Polynomial

The terms polynomials are defined as the parts of the expression that are separated by the operators “+” or “-“. For example, the polynomial expression 2x3– 4x2+ 7x – 4 consists of four terms.

Like Terms and Unlike Terms

It is defined as, Like terms in polynomials are those terms which have the same variable and same power, whereas terms that have different variables and different powers are known as, unlike terms. Suppose, if a polynomial has two variables, then all the same powers of anyone variable will be known as like terms. Let us understand with the help of two examples given below:
2x and 3x are like terms. Whereas, 3y4 and 2x3 are unlike terms.

Degree of a Polynomial

The degree of a polynomial is defined as the highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. The degree is used to determine the maximum number of solutions of a polynomial equation, it means the polynomial equation has one variable which has the largest exponent.
Example 1: A polynomial equation is 3x4+ 7. So it has a degree equal to four.
The degree of the polynomial with more than one variable is equal to the sum of the exponents of the variables in it.
Example 2: Find the degree of the polynomial of 3xy.
In the above polynomial, the power of the x and y variable is 1. To calculate the degree in a polynomial with more than one variable, we can add the powers of all the variables in a term. Then, we get the degree of the given polynomial (3xy) is 2.
Similarly, we can find the degree of the polynomial 2x22y4+7x2y by finding the degree of each term.
So, the highest degree will be the degree of the polynomial. So the given example degree of the polynomial is 6.

What are the Rules for an Expression to be a Polynomial?

The rules of expression to be a polynomial is as given below. An expression should not consist of,

  • The square root of variables.
  • Fractional powers on the variables.
  • Negative powers on the variables.
  • Variables in the denominators of any fractions.
  • A few, not Polynomial examples are 6x-2 is not a polynomial since there is negative power on the variable, x−−√ is not a polynomial, because the variable inside is a radical, 1x22 is not a polynomial, because the variable is in the denominator.

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Types of Polynomials

Based on their degree and their power, the polynomials can be categorized. Based on the number of terms, there are mainly three types of polynomials that are listed below:

    1. Monomials
    2. Binomials
    3. Trinomials

Monomial: An algebraic expression that consists of only a single term is known as a monomial. If the single term is a non-zero then only the algebraic expression is known as a monomial. Here, are a few examples of x, -5xy, and 6y22.

Binomial: A algebraic expression that consists of two terms is known as a binomial. It is generally represented as a sum or the difference of two or more monomials. A few examples of binomials are x + 5, y2+ 5, and 3x3-7.

Trinomial: While a Trinomial is a type of polynomial, it has three terms. The examples of trinomial is 3x3+8x-5, x + y + z, and 3x + y – 5.

However, based on the degree of the polynomial, polynomials can be further classified into 4 major types, namely:

  1. Zero Polynomial
  2. Constant polynomial
  3. Linear polynomial
  4. Quadratic polynomial
  5. Cubic polynomial

Zero Polynomial: In a given polynomial all the coefficients are equal to zero, then the degree of the zero polynomial is either set equal to -1 or is undefined is called Zero Polynomial.

Constant Polynomial: A constant polynomial contains no variables and the value of the polynomial does not change as there are no variables. In this polynomial, the power of the variable is equal to zero. Any constant that can be expressed with a variable with its exponential power equal to zero is known as a constant polynomial.

Linear Polynomial: In this, a Polynomials with 1 as the degree of the polynomial is called a linear polynomial. An example of a linear polynomial is x + y – 4.

Quadratic Polynomial: A Polynomials with 2 as the degree of the polynomial is called quadratic polynomials. The example is 2p2 – 7.

Cubic Polynomial: It is defined as a Polynomials with 3 as the degree of the polynomial is called cubic polynomials. The cubic polynomial example is 6m3-mn+ n2 – 4.

Properties of Polynomials

A polynomial expression is terms connected by the addition or subtraction operators. Based on the type of polynomial and the operation performance we have different properties and theorems. Some of the properties are as given below,

Property 1: Using Division Algorithm
If a polynomial P(x) is divided by a polynomial G(x) the result in quotient Q(x) with remainder R(x), then P(x) = G(x) . Q(x)+R(x)

Property 2: Bezouts Theorem
The Polynomial P(x) is divisible by binomial x-a, if and only if P(a) = 0.

Property 3: Remainder Theorem
The property states that, if P(x) is divided by (x – a) with remainder r, then P(a) = r.

Property 4: Factor Theorem
A polynomial P(x) is divided by Q(x) resulting in R(x) with zero remainders if and only if Q(x) is a factor of P(x).

Property 5: Intermediate Value Theorem
If P(x) is a polynomial, P(x) ≠ P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].

Property 6:
The addition, subtraction, and multiplication of polynomials P and Q result in a polynomial is,
Degree(P ± Q) ≤ max(Degree P, Degree Q) with the equality if degree P ≠ deg ⁡Q
Degree(P × Q) = Degree(P) + Degree(Q)

Property 7:
It states that, if a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.

Property 8:
This property states that, if a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by (Q • R).

Property 9: 
The Polynomial P(x) of degree n>0 has a unique representation of the form. If P(x) = a0 + a1x + a2x2 + …… + anxn is a polynomial where x1,… xn are complex numbers, not necessarily distinct. such that deg(P) = n ≥ 0 then, P has at most n distinct roots or n different zeros.

Property 10: Descartes’ Rule of Sign
The number of positive real zeroes in a polynomial function P(x) will be the same or less than by an even number as the number of changes in the sign of the coefficients. So, if there are ‘k’ sign changes, the number of roots will be ‘k’ or (k – a)’, where ‘a is some even number.

Property 11: Fundamental Theorem of Algebra
This property states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Property 12:
If P(x) is a polynomial with real coefficients and has one complex zero (x = a – bi), then x = a + bi will also be a zero of P(x). Also, x2 – 2ax + a2 + b2 will be a factor of P(x).

Operations on Polynomials

The basic algebraic operations will be performed on polynomials of different types. They are four basic operations on polynomials. The operations are:

  1. Addition of polynomials
  2. Subtraction of polynomials
  3. Multiplication of polynomials
  4. Division of polynomials

Addition of Polynomials: It is one of the basic operations, we can use to increase or decrease the value of polynomials. Whether to add numbers together or to add polynomials, the basic rules remain the same. The only difference is that when you are adding 34 to 127, you align the appropriate place values and carry the operation out. However, when dealing with the addition of polynomials, one needs to pair up like terms and then add them up. Otherwise, all the rules of addition from numbers are translated to polynomials.

Subtraction of Polynomials: The rules for the subtraction of polynomials are very similar to subtracting two numbers. We just align the given polynomials based on the like terms.

Multiplication of Polynomials: The multiplication operation on polynomials follows the general properties like commutative property, associative property, distributive property, etc. Applying these properties using the rules of exponents we can solve the multiplication of polynomials. For example,
(2x + 3y)(4x – 5y) = 2x(4x – 5y) + 3y(4x – 5y) = 8x2 – 10xy + 12xy – 15y2 is 8x2 + 2xy – 15y2.

Division of Polynomials: The division of polynomials is an arithmetic operation where we divide a given polynomial by another polynomial which is generally of a lesser degree in comparison to the degree of the dividend.

Factorization of Polynomials

The Factorization of polynomials is the process by which we decompose a polynomial expression into the form of the product of its irreducible factors, such that the coefficients of the factors are in the same domain as that of the main polynomial. There are different techniques that can be followed for factoring polynomials are:

  1. Method of Common Factors
  2. Grouping Method
  3. Factoring by splitting terms
  4. Factoring Using Algebraic Identities

Based on the complexity of the given polynomial expression, we can use any of the above-given methods.

Polynomial Equations

A polynomial is an equation formed with variables, exponents, and coefficients together with operations and an equal sign. The general form of a polynomial equation is P(x) = an xn + . . + rx + s. Some examples of polynomial equations are x2 + 3x + 2 = 0, x3 + x + 1 = 0, x + 7 = 0, etc.

Polynomial Functions

The general expressions containing variables of varying degrees, coefficients, positive exponents, and constants are known as polynomial functions. Here are some example of polynomial functions, f(x) = x2 + 4, g(x) = -2x3 + x – 7, h(x) = 5x4 + x3 + 2x2

Solving Polynomials

We can solve any polynomial by using factorization and the basic concepts of algebra. Solving a polynomial means finding the roots or zeros of the polynomials. We can apply different methods to solve a polynomial depending upon the type of the polynomial, whether it is a linear polynomial or quadratic polynomial. The first rule for solving a polynomial is to set the right-hand side of the polynomial will be 0.

1. Solving Linear Polynomials: The steps to solve linear polynomial is, Equate the given equation with 0, Make the equation equal to zero, then solve the equation, using basic concepts of algebra.
2. Solving Quadratic Polynomials: The steps to solve quadratic polynomial is, first, we need to rewrite the given expression in the ascending order of degree. Next, Equate the equation with zero and then use the factorization method to solve the equation.

Zeros of Polynomials

The roots or zeros of the polynomial are the real values of the variable for which the value of the polynomial would become equal to zero. If we say any two real numbers, ‘α’ and ‘ß’ are zeroes of polynomial p(x) that is p(α) = 0 and p(ß) = 0. For example, for a polynomial, p(x) = x2 – 2x + 1, we observe, p(1) = (1)2 – 2(1) + 1 = 0. Therefore, 1 will be a zero or root of the given polynomial. It also means the (x – 1) is a factor of p(x). Now, to find the zero or root of any polynomial is that to solve any polynomial, we can apply different methods,

  • Factorization
  • Graphical Method
  • Hit and Trial Method

FAQs on Polynomial

1. What is meant by a Polynomial and not a Polynomial? 

If an algebraic expression consists of a radical then it is not a polynomial. In order for a polynomial, all the exponents present in the algebraic expression should be non-negative integers.

2. What is the standard form of the polynomial?
A standard polynomial is the one where the highest degree is the first term, and subsequently, the other terms come. For example, x3 – 3x2 + x -12 is a standard polynomial. So the highest degree here is 3, then comes 2, and then 1.

3. What makes something not polynomial?
All the exponents in the algebraic expression must be non-negative integers in order for the algebraic expression to be a polynomial. Based on the general rule of thumb if an algebraic expression has a radical in it then it is not a polynomial.

4. What are the rules of polynomials?
In particular, for an expression to be a polynomial term, it must contain no square roots of variables, no fractional or negative powers on the variables, and no variables in the denominators of any fractions.

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