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, \(P(x) = 3x^4 + 2x^2 - x + 7\). 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 \(\deg(3x^4 + 2x^2 - x + 7) = 4\) is 4.


 

Before finding the degree of a polynomial, it helps to understand the difference between monomials and polynomials.
 

• Monomial: A single term with numbers and/or variables multiplied together. For example, \(5x^3 \quad \text{or} \quad 2x y^2 2xy^2\)
  
   
• Polynomial: A sum or difference of one or more monomials. For example,  \(3x^2 + 2x - 7 \quad \text{or} \quad 2x^2y + 3xy^3 - 4x\)
   

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, the degree of the zero polynomial is undefined because there is no non-zero term with the highest power.

A constant polynomial is a polynomial that contains only constant terms without any 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.

For polynomials with more than one variable, the degree of a term is the sum of the exponents of all variables in that term. The degree of the polynomial is the largest degree among all its terms. 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

 

• Degree of 10xy = 1 + 1 = 2
  
   
• Degree of 5x²y³ = 2 + 3 = 5
  
   
• Degree of −2x⁴ = 4 + 0 = 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 degree of a polynomial tells us the highest power of its variable(s) and helps understand the polynomial’s behavior. By following a few simple tips and tricks, you can quickly determine the degree correctly and avoid common mistakes.

 

1. Don’t just look at the first term. Compare the powers of all terms to find the highest.
   
    
2. Simplify the polynomial by combining like terms before finding the degree.
   
    
3. Only non-negative integers count for the degree of a polynomial. Ignore terms with negative or fractional powers.
   
    
4. Add exponents of all variables in a term to find its degree; the highest sum is the degree.
   
    
5. A nonzero constant (like 7) has degree 0; if all terms are zero, the degree is undefined.

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

• Robotics: Polynomials model robot motion. Linear polynomials create straight paths, quadratic enable curves, and cubic or higher degrees allow precise multi-joint movements, ensuring smooth, accurate, and efficient robot actions.

• Business and Economics: Polynomials are used to model profit, revenue, and costs. For example, a quadratic function can represent cost-profit relationships. The degree determines how revenue or profit changes as sales increase, helping businesses forecast and plan.

• Computer Graphics and Animation: Polynomials generate smooth motion in animations and games. The degree controls movement shapes, linear for straight motion, quadratic for parabolas like jumps, cubic or higher for complex curves and realistic motion.

• Machine Learning and Data Science: Polynomial regression fits data trends for predictions. Linear polynomials model simple relationships, quadratic polynomials capture curves, and cubic or higher degrees capture more complex patterns. Choosing the correct degree improves prediction accuracy.
  
   
• Engineering and Physics: Polynomials model trajectories, forces, and energy in structures, projectiles, and machines. The degree indicates how the system changes, higher-degree polynomials model complex motions or forces more accurately.