Degree of Polynomial

Since the degree is the largest exponent on a variable, we look at the powers to identify the degree. For example, if the degree of a polynomial is 5, then the equation will look like this: 

3x⁵ + 2x³ - 8x -3

Here, we don’t look at the number before the variable to find the degree, only the exponents.

Remember that the degree of the polynomial refers to the highest power of one of the variables. We should not confuse variables with constants while finding the degree.

To find the degree of a polynomial using the example, P(x) = 3x⁴ + 2x² - x + 7. In the above example, the degree of the polynomial is 4. We can represent the degree of the polynomial as deg(p(x)). Therefore, the deg(3x⁴ + 2x² - x + 7) is 4.

The polynomial where all the coefficients are zero is called a zero polynomial. It can be written as f(x) = 0. 

We can write it as:

f(x) = 0 × x⁰, 

f(x) = 0 × x¹,

f(x) = 0 × x²,

f(x) = 0 × x³, and so on. 

No matter how much we write, multiplying any number becomes zero, and the degree of the zero polynomial is undefined.

A constant polynomial is a polynomial that has only numbers and not variables. Since the variable x is not present, the value of the polynomial remains the same. We can write it as p(x) = c, where c is just a number like 10, 12, 5, etc.

We can also imagine it as p(x) = c × x⁰, because x⁰ is 1; therefore, multiplying 1 by any number gives the same number. For example, if p(x) is 8, we can also write it as P(x) = 8x⁰. Thus, a constant polynomial always has a degree of 0.

If the polynomials have more than one variable, then the degree is calculated by adding the exponent of each variable. Let us understand more about the polynomial with more than one variable using the following example. 

Calculate the degree of polynomial 10xy + 5 x²y³ - 2x⁴

To find the degree of a polynomial with more than one variable, we need to add the powers of both variables. 

The degree of 10xy is 2, as x and y consist of power 1.

The degree of 5 x²y³ is 5, here, add the power of x(2) and y(3).

The degree of 2x⁴ is 4.

Therefore, the degree of the polynomial is 5.

Polynomials are named based on the highest power of the variable. Given below are some of those polynomials:

The real-life applications of degree polynomials show different fields where polynomials are used and how the degree matters in those situations. 

• Healthcare: In healthcare, polynomials are used to model things like the growth of tumors, the spread of disease, or how a medicine affects the body over time. The degree of polynomials helps doctors to understand whether the growth is slow, steady or speeding up.

• Business and Economics: In business, linear or quadratic polynomials are used to calculate profit, costs, and revenue. The degree helps in predicting future profits and making better business decisions. 

• Computer graphics and Animation: While creating animations or video game characters that move and change shape, polynomials are used to make smooth motion. The degree controls the shape of motion, like jump, bounce, or curve.

• Machine Learning and Data Science: Polynomial equations are used to fit the data in a curve, which is called regression in machine learning and data science. Here, linear polynomials are used for simple trends, and quadratic and cubic polynomials are used for more complex patterns. The right degree predicts more accurate decisions.