What is a Polynomial? 

Polynomial is an algebraic expression with a non-negative integer exponent for variables. Suppose that there are 10 students in a class. If each one is given 2 pencils, then what is the total number of pencils the entire class has? To find the total value we can either add 2 ten times or multiply 2 and 10. Suppose 2 new students join the class. Now the total number of pencils is 24. If 5 more students join, the total number of pencils is 34. The number of students in each case is varying. So, we can mention it generally as 2x where x is the number of students. Here x is called variable and 2 is a constant and since it is multiplied with x it is called the coefficient of x. A term that does not change is called a constant. In 2x+4 , 4 is constant. 2x+4 is an algebraic expression, and it can be added or multiplied. Some terms like -x, 10x, 1 2 x are also algebraic expressions. Polynomial is one of these kinds.

A Polynomial with One Variable

When the number of students is increasing or decreasing, we represent it by the letter x, calling it a variable. Even constants can be represented by any letter. Instead of writing 2x, it can be written as ‘ax’ where a is the constant. Here ‘2’ is called the coefficient of the variable x. Here, ‘a’ will not change, and hence both the letters have two different values and meanings. The perimeter of a square is 4 when multiplied by the length. If the length is 7, the perimeter is 28. The length can be anything. In general, the perimeter is 4x where x can take any positive value in this case. But generally, x can take any real value.

"Polynomial with one variable"


Now let us see the area of a square. The value is found out by multiplying length with length. That is, x×x= x 2 . In algebra, the number two means that the variable x is multiplied twice. If x is multiplied thrice, it is denoted as x 3 . Here, 2 and 3 are called exponents of the variable x. The exponents should always be a whole number for polynomials. Here, x 2 is of degree 2 since the power of the variable x is 2 and x 3 is of degree 3.


Constant Polynomial 

The term 2 is called a constant polynomial. Here the variable x has degree 0 and the value of x 0 is 1. It can also be written as 2 x 0 . Similarly, -3, 8, 19 are also called constant polynomials. 0 is called a zero polynomial.Is x 1 , x 3 , x 3 5 a polynomial? No, because the exponents are not whole numbers.

Degree of a Polynomial

Consider the expression x 2 +3x+1 . This is a polynomial because the exponents are whole numbers. There are 3 terms added up. They are x23x and 1. The degree of x 2 is 2, the degree of 3 x is 1, and the degree of 1 is 0 (1 can be written as 1 x 0 ). The highest degree is the exponent of x with the greatest whole number of all. Here it is 2. Here, 2 is the degree of the polynomial. Now find out for expression, x 4 +9 x 3 +2 . The answer is 4. Look at the previous steps and find out how it is 4. If the degree of the polynomial is two, it is called quadratic. For example, a x 2 +bx+c , where a, b and c are constants, and they can be any real number. The constant a cannot be zero because if a is zero, then the first term vanishes (zero multiplied with anything is again zero). So the highest degree is not 2 anymore. Then it is not quadratic. So, it is expressed as a0 for a quadratic polynomial.

"Degree of a polynomial"

If the highest degree is 3, it is called cubic. For example, a x 3 +b x 2 +cx+d, where the constant a cannot be 0. Why? It is just like the above case.

Zeroes or Roots of a Polynomial 

Let us try substituting different values for the variable x in p(x)=x+3.

When x=1p(1)=1+3=4.

When x=0p(x)=0+3=3.

When x=-1p(x)=-1+3=2.

What is p(x) at -3?



-3 is the zero or root of p(x)=x+3.

Now can you find the root of p(x)=x-9?

(The answer is 9).

Let's now try substituting 0 in the place of the variable x in the expression x2+3x+2.

This is a quadratic polynomial p(x) and can be written as


Replacing x with 0,




The result is 2. Can you find a number which on substituting in the place of x equates p(x) to 0?

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check. 





So, at x=-2, the equation is 0. This is called zero or root of the equation p(x).

Can you find any other number other than -2? What about -1? It also equates p(x) to 0. This is also a root for the equation p(x). Can there be any more roots for x 2 +3x+2 other than -2 and -1? No. Because the degree is 2, only 2 roots can be derived.

Try to find the zeroes of p(x) where p(x)= x 2 +2x15 .

Hint: Substitute -5 and 3 in the place of the variable x

But this is a very long process as you may have to substitute many values and wait for 0 to arrive. Try equating the polynomial to zero and find the root.

If p x = x11 , equating it to 0 gives x-11=0 Now add 11 on both sides. It is x=11.

So, 11 is the root of p x = x11 .

For quadratic polynomials, use the factorization method. In p(x)= x 2 +3x+2 , the coefficient of the middle term x is 3.

Write 3 as the sum of 2 numbers in a way that when you multiply the two numbers you should get the constant term 2.

The two numbers are 2 and 1 because when you multiply 2 and 1 you will get 2 (the constant term p(x)).

So, x 2 +3x+2 is written as x 2 +2x+x+2 .


x 2 +2x+x+2=x(x+2)+(x+2) = (x+1)(x+2).

Equating this to 0 we get, x+1 x+2 =0.

x+1 x+2 becomes 0 when x=1 or x=2 . -1 and -2 are the roots.

For the middle terms which cannot be written as the sum of 2 numbers where on multiplication it equals to the constant term, we can use the formula for finding the roots of the quadratic equation:

b+ b 2 4ac 2a , b b 2 4ac 2a

Here a is the coefficient of x 2 , b is the coefficient of x and c is the constant term.


The general form of a polynomial of degree n in one variable is

p x = a n x n +  a n1 x n1 ++  a 1 x+  a 0

Where n is a non-negative integer, a0, a1, an-1, and an are real numbers, and an0.

The quadratic formula to find zeroes of quadratic polynomials of the form

p x =a x 2 +bx+c (a0) is

b+ b 2 4ac 2a , b b 2 4ac 2a

Where, a, b, and c, are real numbers such that a is non-zero.

Context and Applications 

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for:    

  • Bachelor of Science in Mathematics        
  • Master of Science in Mathematics 

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