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.
- 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.
(i) 3x + 5 is a polynomial in x of degree 1.
(ii) 4y2 -3y +8 is a polynomial in x of degree 2.
(iii) 2u3 + 4u2 -5u + √3 is a polynomial in u of degree 3.
(iv)√(x+3), 1/(x+2), etc, are not polynomials
Polynomial or Not?
These are polynomials:
- x − 4
- −4y2 − (79)x
- 4xyz + 3xy2z − 0.1xz − 200y + 0.5
- 512v5 + 99w5
(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 + 2xn-1 + 5xn-2 } – 3)
|Constant or Zero Polynomial||0||10|
Example: Find the degree of the polynomial 4u4 – 5u3+ 6u2 -8u+3
The degree of the polynomial is 4.
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.
Polynomials can be multiplied using the distributive property.
For example, (x + 2)(x – 3) = x2 – x – 6.
Polynomials can be divided using long division or synthetic division.
For example, (x2 + 2x + 1) ÷ (x + 1) = x + 1.
Polynomials can be evaluated for a specific value of the variable.
For example, P(x) = x2 + 2x + 1, P(2) = 9.
Polynomials can be factored into their constituent factors.
For example, x2 + 3x + 2 can be factored as (x + 1)(x + 2).
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.
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: