Polynomial in One Variable
The degree of polynomials in one variable is the highest power of the variable in the algebraic expression. For example, in the following equation: x2+2x+4. The degree of the equation is 2 .i.e. the highest power of variable in the equation.
For a multivariable polynomial, it the highest sum of powers of different variables in any of the terms in the expression. Take following example, x5+3x4y+2xy3+4y2-2y+1. It is a multivariable polynomial in x and y, and the degree of the polynomial is 5 – as you can see the degree in the terms x5 is 5, x4y it is also 5 (4+1) and so the highest degree among these individual terms is 5.
A polynomial of two variable x and y, like axrys is the algebraic sum of several terms of the prior mentioned form, where r and s are possible integers. Here, the degree of the polynomial is r+s where r and s are whole numbers.
Note: Exponents of variables of a polynomial .i.e. degree of polynomials should be whole numbers.
How to find the Degree of a Polynomial?
There are 4 simple steps are present to find the degree of a polynomial:-
Classification Based on the Degree of the Equation
Based on the degree, the equation can be linear, quadratic, cubic, and bi-quadratic, and the list goes on.
Importance of Degree of polynomial
Case of Homogeneous Polynomial
The degree of terms is a major deciding factor whether an equation is homogeneous or not. A polynomial of more that one variable is said to be homogeneous if the degree of each term is the same. For example, 2x7+5x5y2-3x4y3+4x2y5 is a homogeneous polynomial of degree 7 in x and y.
Relation of Degree of Polynomials with Zeroes of Equation
Theorem 1: A polynomial f(x) of the nth degree cannot vanish for more than n values of x unless all its coefficients are zero.
The above table shows possible real zeros /solutions; actual real solutions can be less than the degree of the equation.
Note: A constant polynomial is that whose value remains the same. It contains no variables. The power of the constant polynomial is Zero. Well, you can write any constant with a variable having an exponential power of zero. If the constant term = 4, then the polynomial form is given by f(x)= 4x0
Before going to start other sections of Polynomials, try to solve the below-given question.
A Question for You
Question: Find the degree of polynomial x3+4x5+5x4+2x2+x+5.
Degree of equation is the highest power of x in the given equation .i.e. 5.