Polynomials

Polynomials Overview

The meaning of Polynomials is ‘many terms’, and it consists of coefficients and variables. These coefficients can be added, subtracted, or multiplied for various mathematical operations. While this chapter imparts knowledge about important terms like factoring polynomials, studying through these notes will help you learn the concept right.

You will be able to solve the exercise questions and answer them correctly once you have thoroughly read these notes. Now ensure that your academic performance gets better with these quality notes covering the intricacies of Polynomials. 


Introducing Polynomials 

Every polynomial is said to have a constant, a variable, and an exponent. It may have more than one terms and the number of terms determine the type of polynomial it is. For instance, take x2 + 5x + 3 as a polynomial expression. Clearly, it has 3 terms and hence can be called a trinomial. Monomial, binomial, etc. are few other kinds of polynomials here. 

In case polynomials are classified depending upon their degree, they are segregated into – 

  • Linear – Expressions having degree as 1. 

  • Cubic – Expressions having degree as 3.

  • Quadratic – Expressions having degree as 2.


Examples of Polynomial

  1. x + y 

  2. 25 

  3. 2x + y + 5 

  4. a + b + c + d 

  5. x2 + x + 2 

  6. x3 + y2 + 2x + 2 

The algebraic expression for writing polynomials is as follows – 

p (x) = a0xn + a1xn-1 + a2xn-2 + … an 

Where, a0, a1, … … ... an denotes the real numbers and the value of n is a positive integer. 


Factor Theorem

Consider a polynomial p (x) with degree equal to or greater than one, where ‘a’ is any real number. Then, we can conclude, 

  1. If p (a) is ‘0’, then (x – a) will be a factor of p (x). 

  2. If (x – a) factorises p (x), then p (a) will be 0. 


Remainder Theorem 

Consider a polynomial q (x) with degree equal to or greater than one, where ‘a’ is any real number. Then, we can conclude, dividing polynomials q (x) by a linear polynomial (x – a), then its remainder should be q (a). 


Adding and Subtracting Polynomials 

You can also add or subtract polynomials. To do so, you must add the like terms together or subtract from like terms. 

For instance, take two polynomials, as shown below. 

3 x2 + 5x + 8, 

and 2 x2 – x – 2. 

Place the like terms together and proceed to add. 

3 x2 + 2 x2 + 5x – x + 8 – 2 

Add the like terms together to get

(3 + 2) x2 + (5 – 1) x + (8 – 2) 

5 x2 + 4 x + 6 

Similarly, you can add or subtract polynomial terms by placing the like terms together and adding them. 

In case of subtraction, consider these polynomials 3 x2 + 5x + 8 and 2 x2 – x – 2. 

Place the like terms together and proceed to subtract. 

3 x2 – 2 x2 + 5x + x + 8 + 2 

Add the like terms together to get this 

(3 – 2) x2 + (5 + 1) x + (8 + 2) 

x2 + 6 x + 10

Now that you are familiar with the idea of multiplying polynomials, you will be able to solve the exercise questions effortlessly. It is critical to learn the theoretical concept and the method so that you can solve mathematical questions quickly.

The quality notes prepared by our expert tutors are meant to help you learn the concepts in an easy manner. Now start preparing for your upcoming exam with our notes and always score high grades in the exam. Now you can also download our Vedantu app for easier access to these materials.

FAQ (Frequently Asked Questions)

1. How to Differentiate if One is a Polynomial or Not?

Ans. A polynomial is comprised of variables, exponents, constants, and these can be added, subtracted, multiplied, or divided. However, there are certain exceptions or rules that states which equations or expressions cannot be a polynomial. If there is a variable in the denominator or the expression is divided by the variable, then it is not a polynomial. It is impossible for polynomials to have negative or fractional exponents. Further, an expression containing radical is not a polynomial as they cannot have radicals. These rules apply in differentiating between a polynomial and non-polynomial.

2. How can I Find the Degree of a Polynomial?

Ans. Degree of a polynomial can be defined as the largest exponent value present in any of the variables in that expression. For example, consider a polynomial 7x²y²+5y²x+4x². In this, the first term 7x²y² has 4 in the exponent (acquiring 2 from x² and acquiring another 2 from y²). The second term 5y²x has a degree of 3 (acquiring 2 from y² and 1 from x). Similarly, the third term 4x² has a degree of 2 acquiring from x². Clearly, the highest degree of a polynomial is 4, in this case.

3. What are the Different Kinds of Polynomials?

Ans. There are several types of polynomials present which are named depending upon the number of terms or degree of a polynomial. The different kinds can be monomials, binomials, trinomials, quadrinomials or simply polynomials. Monomials refer to the expressions which have a single term, e.g. 2x, 3y, etc. Binomials are the polynomials with two terms, for e.g. 2x + 3, y – 4, etc. Likewise, trinomials are the expressions with three terms, for e.g. 2x + 5y + 7. Likewise, depending upon the number of terms in the polynomial, there can be several kinds of it.

4. Why is 8 Considered to be a Polynomial?

Ans. A polynomial is an expression containing variables, constants, and exponents, and is of the form A x² + B x + C (In case of a trinomial). Now, 8 can also be represented in this form. We can write 8 as 0 x² + 0 x + 8. Therefore, it fulfils the fundamental criteria of being a polynomial. Besides, any constant can be called a polynomial because of this same reason. So, any numbers 1, 2, 3, 4, etc. can be called as a polynomial.