A polynomial is a mathematical expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication. The constants of the polynomials are real numbers, whereas the exponents of the variables are positive integers.
For example, ` x^2 + 5x-3` is a polynomial in a single variable x.
The highest power of the variable present in a polynomial is called the degree of the polynomial. In the above example, the degree of the polynomial is 2.
The constant values present in a polynomial are knows as its coefficients / coefficient values. The constants used in the above polynomial are 1, 5 and -3.
Variables are alphabets like a,b,c, x, y, z etc that are used in a polynomial. They are called variables because they can take up any value from a given range (thus called "vary-ables"). In the above example, x is the variable. There also exist polynomials that use more than one variable. For example,` x^2 + 5xy-3y^2 ` is a polynomial of degree 2 in two variables x and y.
Until this point, we have only mentioned what a polynomial is. However, for many reasons it is wise to make clear as to what is not a polynomial. Without this specification, it is likely that you render a non-polynomial to be a polynomial.
Each part of a polynomial that is being added or subtracted is called a "term". So the polynomial has three terms.
Shown above is a simple example of the polynomial, and this is how polynomials are usually expressed. The term having the largest value of exponent (2 in this case) is written first, and is followed by the term with the next lower value of exponent which in turn is followed by a term with the next lower exponent value and so on. The term with the maximum value of exponent is called the "Leading Term" and the value of its exponent is called the "degree of the polynomial".
When a variable is absent in a term, its exponent is zero)
Now the degree of the polynomial is given by the maximum of the above degrees.
=> Degree of the given polynomial = max( 4, 3, 2, 2)
=> Degree of the given polynomial = 4
( Interesting Note: 0 is also a polynomial)
For Example :
Here we need to mention that subtraction, division, matrix multiplication, vector product are all non-commutative.
Similarly Matrix Multiplication and vector Product can be shown to be non-commutative.