Operations on Polynomials
There are four major polynomial operations:
- Addition of Polynomials
- Subtraction of Polynomials
- Multiplication of Polynomials
- Division of Polynomials
- Factorization of Polynomial
Addition of Polynomials
When adding polynomials, it is important to combine the like terms, which means adding the terms that have the same variable and exponent. Adding polynomials will always yield a polynomial of the same degree as the original polynomials being added.
Example: Add the polynomials 3x2 + 2x + 1 and 2x2 – 4x + 3.
Solution:
Identify like terms,
- x2 are 3x2 and 2x2 are like terms.
- x are 2x and -4x are like terms.
- Constant terms are 1 and 3.
Combining the like terms, we have and add the coefficients of the like terms
(3x2 + 2x2) + (2x – 4x) + (1 + 3)
Simplifying,
5x2 – 2x + 4
Therefore,
(3x2 + 2x + 1) + (2x2 – 4x + 3)
= 5x2 – 2x + 4
Subtraction of Polynomials
When it comes to subtracting polynomials, the process is similar to addition, but with a different operation. You subtract the like terms to find the solution. It’s important to remember that subtracting polynomials will always result in a polynomial of the same degree.
Example: Subtraction the polynomial 2x2 + 3x – 5 from the polynomial 4x2 – 2x + 7.
Solution:
(4x2 – 2x + 7) – (2x2 + 3x – 5)
= 4x2 – 2x + 7 – 2x2 – 3x + 5
= (4x2 – 2x2) + (-2x – 3x) + (7 + 5)
= 2x2 – 5x + 12
Learn More: Addition and Subtraction of Polynomials
Multiplication of Polynomials
When two or more polynomials are multiplied together, the resulting polynomial will generally have a higher degree than the original polynomials, unless one of them is a constant polynomial.
Read More: Multiplying Polynomials
Example: Multiply the polynomials (x + 2) and (x – 3).
Solution:
= (x + 2) × (x – 3)
= x × x + x × (-3) + 2 × x + 2 × (-3)
= x2 – 3x + 2x – 6
= x2 – x – 6
Division of Polynomials
It is an arithmetic operation by which a polynomial is divided by another polynomial in a known as polynomial division. For this operation to succeed, the divisor polynomial must have a degree that is less than or equal to the dividend polynomial.
There are several ways to divide polynomials, some of which include:
- Long Division
- Synthetic Division
- Polynomial Division Using Factors
Read more about Dividing Polynomials
Polynomials – Definition, Standard Form, Types, Identities, Zeroes
Polynomials: In mathematics, polynomials are mathematical expressions consisting of indeterminates (also called variables) and coefficients, that involve only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. They are used in various fields of mathematics, astronomy, economics, etc. There are various examples of the polynomials such as 2x + 3, x2 + 4x + 5, etc.
In this article, we will learn about, Polynomials, Degrees of Polynomials, Examples of Polynomials, Zeros of Polynomials, Polynomial Equations, and others in detail.
Table of Content
- What are Polynomials?
- Polynomials Definition
- Polynomials Examples
- Characteristics of Polynomials
- Standard Form of a Polynomial
- Degree of a Polynomial
- Degree of Single Variable Polynomial
- Degree of a Multivariable Polynomial
- Terms in a Polynomial
- Types of Polynomials
- Properties of Polynomials (Theorems of Polynomials)
- Operations on Polynomials
- Addition of Polynomials
- Subtraction of Polynomials
- Multiplication of Polynomials
- Division of Polynomials
- Factorization of Polynomials
- Methods of Factorization of Polynomial
- Greatest Common Factor (GCF)
- Substitution Method
- Grouping Method
- Difference of Two Squares Identity
- Zeros of Polynomial
- How to Find Zeros of Polynomials?
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial
- Higher Degree Polynomial
- Polynomial Identities
- Polynomial Equations
- Solving Polynomials
- Polynomial Functions
- Polynomials Class 9 Extra Questions
- Polynomials Class 10 Extra Questions
- Practice Problems on Polynomial
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