What is a Polynomial?
An expression of the form p(x) = a0 + a1x + a2x2 +….+anxn, where an ≠ 0 is called a polynomial in x of degree n.
Here a0, a1, a2,….., an are real numbers and each power ox x is a non-negative integer. A polynomial can be
- constants (like 5, −10, or ½)
- variables (like x and y)
- exponents (like the 3 in x3 ), but only 0, 1, 2, 3, … etc are allowed
Examples:
(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:
- 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 defined as the largest degree of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the highest exponent is called a degree of the polynomial.
| Polynomial | Degree | Example |
|---|---|---|
| Constant or Zero Polynomial | 0 | 10 |
| Linear Polynomial | 1 | 5x+3 |
| Quadratic Polynomial | 2 | 5x2+5x+1 |
| Cubic Polynomial | 3 | 2u3-3u3+8u+1 |
| Quartic Polynomial | 4 | 2y4+3y3-5y2+9y+1 |
Example: Find the degree of the polynomial 4u4 – 5u3+ 6u2 -8u+3
Solution:
The degree of the polynomial is 4.
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:
- Monomial
- Binomial
- Trinomial
These polynomials can be together using addition, subtraction, multiplication, and division but is never division by a variable. A few examples of Non Polynomials are: 1/x+4, x-5