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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: x^{2}+2x+4. The degree of the equation is 2 .i.e. the highest power of variable in the equation.

Multivariable polynomial

For a multivariable polynomial, it the highest sum of powers of different variables in any of the terms in the expression. Take following example, x^{5}+3x^{4}y+2xy^{3}+4y^{2}-2y+1. It is a multivariable polynomial in x and y, and the degree of the polynomial is 5 â€“ as you can see the degree in the terms x^{5} is 5, x^{4}y 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 ax^{r}y^{s} 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:-

Example: 6x^{5}+8x^{3}+3x^{5}+3x^{2}+4+2x+4

- Step 1: Combine all the like terms that are the terms of the variable terms

(6x^{5}+3x^{5})+8x^{3}+3x^{2}+2x+(4+4) - Step 2: Ignore all the coefficients

x^{5}+x^{3}+x^{2}+x+x^{0} - Step 3: Arrange the variable in descending order of their powers

x^{5}+x^{3}+x^{2}+x+x^{0} - Step 4: The largest power of the variable is the degree of the polynomial

deg(x^{5}+x^{3}+x^{2}+x+x^{0}) = 5

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, 2x^{7}+5x^{5}y^{2}-3x^{4}y^{3}+4x^{2}y^{5} 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)= 4x^{0}

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 x^{3}+4x^{5}+5x^{4}+2x^{2}+x+5.

Solution: x^{3}+4x^{5}+5x^{4}+2x^{2}+x+5

=4x^{5}+5x^{4}+x^{3}+2x^{2}+x+5

=x^{5}+x^{4}+x^{3}+x^{2}+5

Degree of equation is the highest power of x in the given equation .i.e. 5.

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