Table of Contents
What is a Polynomial?
A polynomial is a mathematical expression consisting of one or more terms, each of which is an algebraic product of a constant and one or more variables raised to a positive integer power. The degree of a polynomial is the largest power to which a variable appears. Polynomials are used in mathematics and physics to describe the motion of objects and the properties of physical systems.
Types of Polynomials
A polynomial is an algebraic expression consisting of a sum of one or more terms, each of which is a product of a coefficient and a power of a variable. The coefficients may be real or complex numbers.
A polynomial in one variable x can be written in the form
a 0 xn + a 1 xn-1 + a 2 xn-2 + ··· + a n-1 x + a n
where a 0 , a 1 , a 2 , …, a n are real or complex numbers. The degree of a polynomial is the largest degree of any one of its terms.
A polynomial in two variables x and y can be written in the form
a 0 xn + a 1 xn-1 y + a 2 xn-2 y2 + ··· + a n-1 xn-1 yn-1 + a n xn y
where a 0 , a 1 , a 2 , …, a n are real or complex numbers. The degree of a polynomial is the largest degree of any one of its terms.
A polynomial in three variables x, y, and z can be written in the form
a 0 xn + a 1 xn-1 y + a 2 xn-2 y2 + ··· + a n-1 xn-1 yn-1 + a n
Classification on the basis of Terms
There are two types of classification on the basis of terms:
1.Synonym classification
2.Antonym classification
Synonym classification is the process of grouping together words that have the same meaning. Antonym classification is the process of grouping together words that have opposite meanings.
Classification on the Basis of Degrees of Freedom
A system is said to be classified on the basis of degrees of freedom if the number of degrees of freedom of the system is used to classify the system.
There are six different types of classification on the basis of degrees of freedom:
1. Rigid body
2. Fixed body
3. Free body
4. Hinged body
5. Sliding body
6. Rolling body
Operations On Polynomials Addition
To add two polynomials, add the coefficients of like terms together.
Example:
2x + 3x = 5x
Subtraction
To subtract two polynomials, subtract the coefficients of like terms together.
Example:
3x − 2x = 1x
Multiplication
To multiply two polynomials, multiply the coefficients of like terms together.
Example:
3x ⋅ 2x = 6×2
Division
To divide two polynomials, divide the coefficients of like terms together.
Example:
6×2 ÷ 3x = 2x
Multiplication of Polynomials
Multiplication is the process of repeated addition. When multiplying polynomials, it is easiest to think of it as repeated addition of terms.
For example, if we want to multiply 3×2 + 5x + 2 by 4×3, we can think of it as adding 3×2 + 5x + 2 four times:
3×2 + 5x + 2
4×3
12×5 + 20x + 8
Types Of Multiplication
There are three types of multiplication:
1. Associative: This type of multiplication is when the order of the numbers being multiplied does not affect the result. For example, 3 x 4 x 5 = 3 x (4 x 5) = (3 x 4) x 5.
2. Commutative: This type of multiplication is when the order of the numbers being multiplied does affect the result. For example, 3 x 2 = 2 x 3 = 6.
3. Distributive: This type of multiplication is when a number is multiplied by each of the numbers in a set. For example, (4 x 5) x 6 = 4 x (5 x 6) = (4 x 5) + (4 x 6) = 240.
Multiplication Of Two Monomials
The product of two monomials is a polynomial. The two monomials are multiplied together to produce the polynomial. The coefficients of the polynomial are the products of the coefficients of the monomials.
Multiplying Monomial by Binomial
Multiplying a monomial by a binomial is the same as multiplying the coefficients of the monomial and binomial and then adding the terms.
For example, to multiply 3x by 4x:
3x * 4x = 3x^2 + 4x
Multiplication of Two Binomial – The FOIL Method
To multiply two binomial expressions, use the FOIL method.
The FOIL method stands for:
F irst
O utside
I nside
L ast
For more visit Binomial Theorem Class 11 – Definition, Formula, Properties and FAQs
