Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial, and trinomial. Examples of constants, variables and exponents are as follows:
- Constants. Example: 1, 2, 3, etc.
- Variables. Example: g, h, x, y, etc.
- Exponents: Example: 5 in x5 etc.
Standard Form of a Polynomial
P(x) = anxn + an-1xn-1 +an-2xn-2 + ………………. + a1x + a0
Where an, an-1, an-2, ……………………, a1, a0 are called coefficients of xn, xn-1, xn-2, ….., x and constant term respectively and it should belong to real number (⋲ R).
Notation
The polynomial function is denoted by P(x) where x represents the variable. For example,
P(x) = x2-5x+11
If the variable is denoted by a, then the function will be P(a)
Degree of a Polynomial
The degree of a polynomial is defined as the highest exponent of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.
| Polynomial | Degree | Example |
|---|---|---|
| Zero Polynomial | Not Defined | 6 |
| Constant | 0 | P(x) = 6 |
| Linear Polynomial | 1 | P(x) = 3x+1 |
| Quadratic Polynomial | 2 | P(x) = 4x2+1x+1 |
| Cubic Polynomial | 3 | P(x) = 6x3+4x2+3x+1 |
| Quartic Polynomial | 4 | P(x) = 6x4+3x3+3x2+2x+1 |
Example: Find the degree of the polynomial P(x) = 6s4+ 3x2+ 5x +19
Solution:
The degree of the polynomial is 4 as the highest power of the variable 4.
Terms of a Polynomial
The terms of polynomials are the parts of the expression that are generally separated by “+” or “-” signs. So, each part of a polynomial in an expression is a term. For example, in a polynomial, say, 2x2 + 5 +4, the number of terms will be 3. The classification of a polynomial is done based on the number of terms in it.
| Polynomial | Terms | Degree |
| P(x) = x3-2x2+3x+4 | x3, -2x2, 3x and 4 | 3 |
Types of Polynomials
Depending upon the number of terms, polynomials are divided into the following categories:
- Monomial
- Binomial
- Trinomial
- Polynomial containing 4 terms (Quadronomial)
- Polynomial containing 5 terms (pentanomial ) and so on …
These polynomials can be combined using addition, subtraction, multiplication, and division but is never divided by a variable. A few examples of Non Polynomials are: 1/x+2, x-3
Monomial
A monomial is an expression which contains only one term. For an expression to be a monomial, the single term should be a non-zero term. A few examples of monomials are:
- 5x
- 3
- 6a4
- -3xy
Binomial
A binomial is a polynomial expression which contains exactly two terms. A binomial can be considered as a sum or difference between two or more monomials. A few examples of binomials are:
- – 5x+3,
- 6a4 + 17x
- xy2+xy
Trinomial
A trinomial is an expression which is composed of exactly three terms. A few examples of trinomial expressions are:
- – 8a4+2x+7
- 4x2 + 9x + 7
| Monomial | Binomial | Trinomial |
| One Term | Two terms | Three terms |
| Example: x, 3y, 29, x/2 | Example: x2+x, x3-2x, y+2 | Example: x2+2x+20 |
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