Types of Polynomials

Algebraic expressions are formed by combining variables and constants using addition, subtraction, or multiplication. Polynomials are made up of variables, constants, and exponents.

For example, \(5x^2 + 3x + 2\) is a polynomial, but \(4x^2 - 5^{-1} + 8x^{\frac{3}{2}} \) is not a polynomial, as it has a term with a negative exponent \((5x^{-1}) \) and another with a fractional exponent \(7x\left(\frac{3}{2}\right) \).

In standard form, polynomials are arranged in descending order of the exponents, followed by a constant. A coefficient is a number multiplied by a variable, while a constant is a number with no variable.

Based on the number of terms and their degree, polynomials are classified into different types. These are the major types of polynomials:

In polynomial expressions, the degree of a polynomial is the highest exponent of the variable. For example, in \(3x^3 + 4x^2 - 5x + 5\), the degree of the polynomial is 3. Types of polynomials based on degree are:

• Zero polynomial
   
• Constant polynomial 
   
• Linear polynomial 
   
• Quadratic polynomial 
   
• Cubic polynomial 

Zero polynomial: The polynomials having all coefficients as zero are the zero polynomials. It is usually written as 0. 

Constant polynomial: A constant polynomial is a polynomial with a degree of zero, for example, 4⁰.

Linear polynomial: The linear polynomials are formed in the equation, \(p(x) = ax + b\), where the highest degree of the polynomial is 1. For example, \(6x + 5\).

Quadratic polynomial: Polynomials having the highest degree of 2, are quadratic polynomials. For example, \(4x^2 + 2x + 2\).

Cubic polynomial: The polynomials with the highest degree of 3 are cubic polynomials. For example, \(6x^3 + 12x^2 + 3x + 9\)

Based on the number of terms in polynomial expressions, we can classify them into three types. They are:
 

• Monomial
   
• Binomial
   
• Trinomial

An algebraic expression with only one non-zero term is a monomial. For example, \(5xy^2\), \(4x\), \(5m\), etc. The monomial consists of variables, coefficients, and literal parts. 

In \(5xy^2\), the 5 is the coefficient, the variables are x and y, and \(xy^2\) is the literal part. 

The degree of the monomial is the sum of the exponents of the variables. For instance, in \(5xy^2\), the degree of 3 as the exponent of x and y is 1 and 2.

Using the monomial expression, we can perform addition, subtraction, multiplication, and division. 
 

1. Addition: The monomials can be added together when they have the same literal part, and the sum will also be a monomial expression. For example, \(8xy + 6xy = 14xy\)
    
2. Subtraction: Monomial subtraction involves the subtraction of expressions with the same literal part. For example, \(14xy - 8xy = 6xy\)
    
3. Multiplication: In monomial multiplication, first multiply the coefficients and then add the exponents of the same variables. For example, \(8x^2y × 6xy = 48x^3y^2\)
    
4. Division: When dividing two monomials, we first divide the coefficients and then divide the variables by subtracting the exponents. For example, \(6xy^2 ÷ 2xy = 3y\)

The word bi means two. The algebraic expression with two non-zero terms is binomial. It can be represented as \(ax^m + bx^n \), where a and b are the coefficients, x is the variable, and m and n are the exponents. For example, \(5x^2 + 2y\), where 5 and 2 are the coefficients, x and y are the variables, and 2 and 1 are the exponents. 

Now let’s learn the operations of binomials. Some basic operations on binomials are:

Factorization: Factorization can be done for an equation by using certain formulas. Such as, \(x^2 - y^2 = (x + y)(x - y)\)

Addition: Like terms can be added to each other. For example, \((5x^2 + 6y) + (2x^2 + 3y) = 7x^2 + 9y\)

Subtraction: Like terms can be subtracted to each other. For example, \((7x^2 + 9y) - (2x^2 + 3y) = 7x^2 + 9y - 2x^2 - 3y = 5x^2 + 6y\)

Multiplication: We can multiply these equations in such a way of, \((ax + b)(cx + d) = acx^2 + (ad + bc)x + bd\)

Raising to the nth power: We can perform it by, \((x + y)^2 = x^2 + 2xy + y^2\)

Converting to lower-order binomials: Follow the formula to convert to lower-order binomials, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Trinomials are polynomials that consist of exactly three non-zero terms, with different combinations of variables or exponents. For example, \(3x^3 + 9x^2 + 6x\), where 3, 6, and 9 are the coefficients and x is the variable.

Here are some of the tips and tricks for students to master the types of polynomials.
 

1. Understanding what a polynomial is. A polynomial is an algebraic expression consisting of terms made up of a variable raised to a non-negative integer exponent and multiplied by a coefficient.

   Example: \(3x^4 + 2x^3 - x + 7\)

   In the polynomial,
   
   Terms: \(3x^4, 2x^3, -x, 7\)
   
   Coefficients: 3, 2, -1, 7
   
   Degrees: 4, 3, 1, 0
    

2. Classifying polynomials by degree. The degree of a polynomial is the highest power of the variable. When the degree of the polynomial is 0, it is known as constant. When it is 1, the polynomial is known as linear; when it is 2, it is known as quadratic and so on. Always look at the highest exponent to immediately know the type.
    

3. Classifying polynomials by number of terms. If there is only one term in a polynomial, then it is known as monomial; when there are two terms, it is known as binomial; when there are three terms, it is known as trinomial and so on. Count the terms separated by + or − signs. This is easy to spot once you write the polynomial in standard form.
    

4. Always write in standard form: Arrange terms in descending powers of x.
   
   Example: \(x^2+3x−5\) instead of \(3x−5+x^2\)
    

5. Always practice classification regularly. Take 5 random polynomials and classify by degree and terms.

We discussed the different types of polynomials. Now let's see how we use them in real life. In real life, polynomials are used in the fields of physics, engineering, computer science, etc. 

1. Medicine: polynomial expressions are used to model the drug concentration in the patient's blood over time, which helps to determine optimal dosage schedules.
    
2. Finance: polynomials are used to analyze the trend and predict stock prices. Profit or revenue functions often form a quadratic parabola when maximizing/minimizing outcomes. Calculating total cost when price per item is constant (e.g., 𝐶𝑜𝑠𝑡 = 50x+100).
    
3. Physics: polynomial expressions are used to describe projectile motion, where a quadratic function is used to find the height of an object over time under gravitational acceleration. Here, acceleration under uniform gravity leads to quadratic relationships.
    
4. Computer graphics: Smooth curves and animation paths in 3D modeling can be done using these polynomials. We can use quadratic and higher degree polynomials in such cases. 
    

5. Projectile motion: Objects thrown in the air follow a parabolic path. The parabolic path of the object can be tracked with the help of a quadratic polynomial equation, given to the object.