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Polynomials

Degree of Polynomial

Well, before starting, I would like to tell you that this ‘degree’ has nothing to do with your thermometer’s degree or to your course completion certification. The term ‘degree’ has come to the important part of Mathematics, i.e., Polynomials and is adding an essential meaning to it. So, let’s hit directly to understand the Degree of Polynomials in details.

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Degree of Polynomials

degree of polynomials

Source: Wikihow

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.

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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, 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.

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How to find the Degree of a Polynomial?

There are 4 simple steps are present to find the degree of a polynomial:-

Example: 6x5+8x3+3x5+3x2+4+2x+4

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

         (6x5+3x5)+8x3+3x2+2x+(4+4)

  • Step 2: Ignore all the coefficients

        x5+x3+x2+x+x0

  • Step 3: Arrange the variable in descending order of their powers

       x5+x3+x2+x+x0

  • Step 4: The largest power of the variable is the degree of the polynomial

deg(x5+x3+x2+x+x0) = 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.

Name of the Equation Degree of the Equation
Linear Equation 1
Quadratic Equation 2
Cubic Equation 3
Bi-Quadratic equation 4

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.

Name of the Equation Degree of the Equation Possible Real Solutions
Linear Equation 1 1
Quadratic Equation 2 2
Cubic Equation 3 3
Bi-Quadratic equation 4 4

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 1: Find the degree of polynomial x3+4x5+5x4+2x2+x+5.

Solution: x3+4x5+5x4+2x2+x+5
=4x5+5x4+x3+2x2+x+5
=x5+x4+x3+x2+5
Degree of equation is the highest power of x in the given equation .i.e. 5.

Question 2: Give example of a degree of polynomial?

Answer: An example of degree of polynomial can be 5xy2 that has a degree of 3. This is because x has an exponent of 1, y has 2, so 1+2=3.

Question 3: Explain the degree of polynomial under root 3?

Answer: Under root 3 is a polynomial and its degree is 0. This is because its expression can take place as √3(x^0).

Question 4: Explain the degree of zero polynomial?

Answer: The degree of the zero polynomial has two conditions. The conditions are that it is either left undefined or is defined in a way that it is negative (usually −1 or −∞). Similar to any constant value, one can consider the value 0 as a (constant) polynomial, called the zero polynomial. This polynomial has no nonzero terms and hence it has no degree.

Question 5: Explain what you understand by a third degree polynomial?

Answer: First degree polynomials can also be called as linear polynomials. Furthermore, first degree polynomials refer to lines which are neither vertical nor horizontal. More often, use of letter m takes place as the coefficient of x rather than a, and it represents the slope of the line.

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