Nth Degree Polynomial

The greatest power of the variable in a polynomial is known as its degree. Therefore, any polynomial with the highest power of the variable of n is a nth degree polynomial. Thus, the polynomial is in the following form:
\(P(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_0\)

It is a real-coefficient, nth degree polynomial function with the variable denoted by x and the highest power n.

For example, expressions such as √x+2x+7 are not polynomials because the variable x has a fractional exponent, and all the exponents in a polynomial must be whole numbers. The coefficients, such as \(a_n\), \(a_{n-1}\), \(a_{n-2}\), ...., \(a_0\) are the constants that multiply the variable terms. This is also the standard method of representing various types of polynomials, and their coefficients \(a_n\), \(a_{n-1}\), \(a_{n-2}\),...,\(a_0\), and the power n can have numerical values depending on the types of polynomials they represent.
 

The degree of a polynomial in one variable is the highest power of the variable in the polynomial. It is said to be the highest exponent value of the variable in the given polynomial.

Example:
Degree of \((x^4+4x^2+3x)=4\)
Degree of \((x)=1\)
Degree of \((x^2+2x+1)=2\)

What is the Degree of a Polynomial with More than One Variable

When a polynomial contains multiple variables, we find the degree of each term by adding the powers of each variable that is present in the term.

In other words, to find the degree of a polynomial having more than one variable, add the exponents in each term, this gives you the degree of that term. After adding exponents, the exponent of the term having the highest exponent is considered as the degree of that polynomial.


For example, \(5x^5+4xy^2+3xy\)
\(5x^5\) is 5, is the degree of the term.
The degree of \(4xy^2\) is 3 (Sum of exponents = 1+2=3)
The term \(3xy\) has a degree of 2 (Sum of exponents =1+1=2)
5 is the highest degree. Therefore, the polynomial’s degree is 5.
 

Combine all like terms (terms with the same variables and exponents) before determining the degree. This simplifies the polynomial. Then the highest exponent determines the degree of the polynomial.

Finding a polynomial’s degree with one variable
 

Step 1: Write the polynomial expression in the correct form. The correct form of a polynomial refers to arranging the terms in descending order of the degrees.


Step 2: Find which term has the variable’s highest power. The coefficient of this term must be non-zero. The leading term in the correct form is said to be the one with the highest power.

Step 3: The polynomial’s degree is the same as the highest power term’s degree.
For example, \(7x^3+5x^2-3x-4x^3+2-3x^3\)
When we combine similar terms, we obtain
\(p(x)=(7x^3-4x^3-3x^3)+5x^2-3x+2\)
\(p(x)=5x^2-3x+2\)... the term \(x^3\) is eliminated due to the 0 coefficient.

The standard form of this expression already exists.
The term with the highest power of x in this case is \(5x^2\)
Therefore, the polynomial’s degree is 2.

Degree of Zero Polynomial

The expression \(p(x)=0\) or the number 0 is typically used to represent the zero polynomial. Non-zero terms do not exist in the zero polynomial. A polynomial that has zero coefficients is referred to as a zero polynomial. There are no terms with coefficients that are not zero. 


Why is the degree of zero polynomial regarded as undefined? We can rewrite \(p(x)=0 as p(x)=0.x^n\)
\(p(x)=0.x^1\)
\(p(x)=0.x^2\)
\(p(x)=0.x^3\)
…
As a zero polynomial doesn’t have any non-zero terms, its degree is undefined.

Zeros of Nth Degree Polynomial

The values of a variable for which the nth degree polynomial equals zero are known as their zeros. There is a maximum of n zeros in a polynomial of degree n. These zeros might be complex or real.

For example,
The zeros of \(x^3-6x^2+11x-6=0\) are: 
\(x=1, 2, 3\)
The zeros of \(x^2+4=0\) are:
\(x=2i,-2i\) (complex)
The zeros of \( x^4=0\) are:
\(x=0\) (with multiplicity 4)
 

Here are some smart tips and tricks to help you master nth-degree polynomials with confidence!

 

• Identify the degree first: Always look for the highest power of the variable (x) in the polynomial. That exponent is the degree, it determines the number of possible roots and the general shape of the graph.
  
   
• Arrange in standard form: Write polynomials in descending order of powers (highest to lowest). This makes it easier to spot the leading term, degree, and coefficient, which are vital for analysis.
  
   
• Use the leading coefficient to predict behavior: The leading coefficient (the coefficient of \(x^n\)) and the degree together determine how the graph behaves at the ends.
  
  If the degree is even: both ends go in the same direction. If odd: ends go in opposite directions. A positive leading coefficient → rises on the right; negative → falls on the right.
  
  
   
• Practice factoring and root finding: Get comfortable factoring lower-degree polynomials (like quadratic and cubic), this helps build intuition for higher degrees. A nth-degree polynomial has at most n roots, though some may be complex or repeated.
  
   
• Visualize the graph: Plot or sketch the polynomial using tools like Desmos or GeoGebra. Visual learning helps you connect algebraic expressions to curve behavior, like turning points, intercepts, and end behavior.

Let us see how nth degree polynomials help in real life.
 

• Motion of projectiles: A cubic polynomial can be used to model the trajectory of a projectile in the presence of air resistance or complex forces. The zeros indicate the point at which the object touches the ground. Physicists use these points to determine safe landing zones, flight duration, and range.
  
   
• Forecasting revenue: Businesses can use cubic functions to model changes in revenue over time. When revenue drops to zero (also known as break-even points or loss periods). Recognizing these helps companies in risk planning, strategy modification, and increasing profitability.
  
   
• Design of amusement parks: A cubic polynomial can be used to model the path of a roller coaster for smooth curves. The entry and exit points are indicated by the zeros. This is used by the designers to guarantee safe riding and seamless transitions.
  
   
• River flow modelling in environmental science: Cubic functions are used by scientists to model river pollution or water flow. The zeros help in locating areas where pollutants are concentrated or where the flow slows to zero. This helps in planning safety measures, pollution prevention, and environmental planning.
  
   
• Design of gears and cams: Engineers model the motion path of cams and gears using cubic polynomials. The polynomial’s zeros, that represent the locations where the part changes direction or the motion resets. This helps to prevent mechanical failures and assures the accurate timing and smooth rotation.