Polynomials: Types and Examples

What is a Polynomial?

A polynomial is a mathematical equation that consists of variables and coefficients. It involves only the operations of addition, subtraction, and multiplication. The variables in a polynomial can have non-negative integer exponents.
For example, the expression 3x² – 2x + 5 is a polynomial.
where,

• 3, -2, and 5 are coefficients,
• x is a variable,
• and 2 and 1 are the exponents.

The highest exponent in this polynomial is 2, so it is called a quadratic polynomial. Polynomials can have any degree, which is the highest exponent of the variable in the polynomial.

Polynomial or Not?

These are polynomials:

• 5x
• x − 4
• −4y² − (79)x
• 4xyz + 3xy²z − 0.1xz − 200y + 0.5
• 512v⁵ + 99w⁵
• 9

(Yes, “9” is a polynomial because it can be just a constant!)

These are not polynomials

• 5xy^-4 is not polynomials, because the exponent is “-4” (exponents can only be allowed 0,1,2,…)
• 2/(x+4) is not, because dividing by a variable is not allowed
• 1/x is not either
• √x is not, because the exponent is “½”.

Degree of a Polynomial

The degree of a polynomial is the highest power (or exponent) of the variable in the polynomial equation.
For example, in the polynomial expression 4u⁴ – 5u³+ 6u² -8u+3, the degree of the polynomial is 4, which is the highest power of the variable u. This is a fourth-degree polynomial.
Here are given below some properties described to the degree of a polynomial:

• The degree of a non-zero constant polynomial is 0.
• The degree of a non-zero linear polynomial in one variable is 1.
• The degree of the sum or difference of two polynomials is the highest degree of the two polynomials.
• The degree of the product of two polynomials is the sum of the degrees of the two polynomials.

Here are some examples of polynomials of different degrees:

• Degree 0: Constant polynomial (e.g. 6)
• Degree 1: Linear polynomial (e.g. 5x – 6)
• Degree 2: Quadratic polynomial (e.g. x² + 2x + 1)
• Degree 3: Cubic polynomial (e.g. 2x³ – 5x² + 3x + 7)
• Degree n: nth degree polynomial (e.g. 4xⁿ + 2x^n-1 + 5x^n-2 } – 3)

Operations on Polynomials

Several operations can be performed on polynomials. These operations are the basis for solving polynomials.
Here are given some of the most common operations on polynomials:

1. Addition and Subtraction

Polynomials can be added and subtracted by combining like terms.

For example, (3x² + 2x + 1) + (2x² – 3x + 2) = 5x² – x + 3.

2. Multiplication

Polynomials can be multiplied using the distributive property.

For example, (x + 2)(x – 3) = x² – x – 6.

3. Division

Polynomials can be divided using long division or synthetic division.

For example, (x² + 2x + 1) ÷ (x + 1) = x + 1.

4. Evaluation

Polynomials can be evaluated for a specific value of the variable.

For example, P(x) = x² + 2x + 1, P(2) = 9.

5. Factoring

Polynomials can be factored into their constituent factors.

For example, x² + 3x + 2 can be factored as (x + 1)(x + 2).

6. Integration

The integral of a polynomial can be found using the power rule.

For example, if P(x) = x³ + 2x² + x, then the indefinite integral of P(x) with respect to x is (1/4)x⁴ + (2/3)x³ + (1/2)x² + C, where C is a constant.

7. Derivatives:

The derivative of a polynomial can be found using the power rule.

For example, if P(x) = x³ + 2x² + x, then P'(x) = 3x² + 4x + 1.

Types of Polynomials

Polynomials are of three separate types and are classified based on the number of terms in it. The three types of polynomials are given below:

1. Monomial
2. Binomial
3. Trinomial