In Maths, Polynomial is a concept which is studied under the Algebra branch.
Polynomials are a particular type of algebraic expressions, when all variables involved in an algebraic
expression has powers with whole numbers only.
To understand algebraic expressions in details, these can be studied in
Algebraic Expression and its Operations
chapter in details.
Before moving to the definition of a polynomial and its various types, let’s go through once the basic terms
that comprise polynomials.
Variables are defined and used in algebra. Its definition states that a symbol which can be used to assign
different numerical values is known as a variable.
Variables are represented by any small case English letters.
x,y,z,p,q,r,s etc.
In Math, a symbol having a fixed value is called a constant.
8, 5, 9, π etc.
A combination of constants and variables connected by some or all basic operations +, -, X, ÷ is called an algebraic expression.
7 + 8x
So, here we have made an algebraic expression connected by + operation by combining a constant 7 and
a constant 8 with variable x.
7 + 8x – 6x^{2}y – $\frac{4}{9}$xy is
another example of algebraic expression.
Various parts of an algebraic expression separated by + or – operations are called terms.
Consider the algebraic expression as 7 + 8x – 6x^{2}y – $\frac{4}{9}$xy
So, the terms in this expression are 7, 8x, 6x^{2}y, –$\frac{4}{9}$xy
An algebraic expression in which variables involved are having non negative integral powers is called a polynomial.
We can learn polynomial with two examples:
Example 1: x^{3} + 2x^{2} + 5x + 7
Variables involved in the expression are only x.
The power of x in each term is:
x^{3}, x has power of 3
2x^{2}, x has power of 2
5x, x has power of 1
7, 7 has power of 0
So, we can see x variable in terms of the expression has powers as 3, 2, 1 and 0; which are non negative
integrals.
Example 2: x^{2}y + xy^{2} + 7y
So, variables involved in the expression are x and y.
The powers of x and y in each term are:
x^{2}y, x has power of 2 and y has 1
xy^{2}, x has power of 1 and y has 2
7y, y has power of 1
So, we can see x and y variables in terms of the expression have powers as 2 or 1 which are non negative
integrals also.
So, that makes them polynomials.
For a polynomial involving one variable, the highest power of the variable is called degree of the polynomial.
3x^{4} + 5x^{3} + 7x^{2} + 8
This polynomial is one variable polynomial i.e. x.
The powers of the x variable are 4, 3 and 2
∴ the highest power is 4.
Hence, the degree of the polynomial is 4.
Polynomials have different types depending upon the degree of polynomial and number of terms involved in the polynomial.
Polynomials are classified and named on the basis of the number of terms it has.
In general, the naming of type of polynomial is written by prefixing the words mono, bi and tri to “nomial”.
Where mono refers to one, bi refers to two and tri refers to three.
The types are monomial, binomial and trinomial.
Let’s have a look at the various types of polynomials with their examples.
A polynomial containing one non zero term is called a monomial.
8x^{5}
In this polynomial, there is only one non zero term i.e. 8x^{5}. Therefore, it is an example
of a
monomial.
-7y^{3}
Here, -7y^{3} is only one non zero term.Therefore, it is also an example of a monomial.
A polynomial containing two non zero terms is called a binomial.
7x^{6}-3x^{4}
In this polynomial, there are two non zero terms i.e. 7x^{6} and 3x^{4}. Therefore,
it is an example of binomial.
A polynomial containing three non zero terms is called a trinomial.
5x^{4} + 3x^{2} – 8
Here we have three non zero terms i.e.
5x^{4}, 3x^{2} and 8. Therefore, it is an example of a trinomial.
A polynomial containing four terms is called a quadrinomial polynomial.
7x^{5} – 3x^{2} + 9x + 5
A polynomial containing five terms is called quintrinomial polynomial.
y^{6} + 8y^{5} + 9y^{4} + 9y^{2} + 7
A polynomial of degree 1 is called a linear polynomial. A linear polynomial in one variable can have at most two
terms.
Linear polynomial in variable x can be in general form of ax + b
x^{1} + 8
A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial in one variable can have at
most three terms.
Quadratic polynomial in variable x can be in general form of ax^{2} + bx + c
5x^{2} + 2x – 7
A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial in one variable can have at most four
terms.
Cubic polynomial in variable x can be in general form of ax^{3} + bx^{2} + cx + d
2x^{3} + 3x^{2} – 5x + 7
A polynomial of degree 4 is called a biquadratic polynomial. A biquadratic polynomial in one variable can have at
most five terms.
Biquadratic polynomial in variable x can be in general form of ax^{4} + bx^{3} + cx^{2
} + dx + e
2x^{4} + 5x^{3} – 3x^{2} + 7x – 4
A polynomial of degree 0 is called a constant polynomial.
4, -7 etc.
Why?
∵ 4 can be written as 4y^{0}, where the degree of this polynomial is zero.
Also, -7 can be written as -7y^{0}, where the degree of this polynomial is zero.
Hence, 4 and -7 are examples of constant polynomials.
A polynomial is said to be zero polynomial if all coefficients are equal to zero.
0x^{5} + 0x^{3} + 0x^{2} + 0x etc.
Degree of zero polynomial is not defined.
Polynomial | Number of terms | Example |
---|---|---|
Monomial | 1 | 5x^{2} |
Binomial | 2 | 2x^{5} + 5x^{3} |
Trinomial | 3 | 9x^{8} + 3x^{5} – 12 |
Quadrinomial polynomial | 4 | 3x^{6} + x^{4} – 6x – 1 |
Quintrinomial polynomial | 5 | 3x^{9} – 2y^{7} – y^{4} – 2y^{2} – 4 |
Polynomial | Standard form | Degree |
---|---|---|
Constant | ax^{0} | 0 |
Linear | ax + b | 1 |
Quadratic | ax^{2} + bx + c | 2 |
Cubic | ax^{3} + bx^{2} + cx + d | 3 |
Biquadratic | ax^{4} + bx^{3} + cx^{2} + dx + e | 4 |
An algebraic expression in which variables have only whole numbers as its exponents, is called polynomial.
In polynomial, the highest power of variable is called degree of polynomial.
The degree of constant polynomial is always zero.
Yes, because we can write 100 as 100x^{0}, so the highest degree is zero, which is called as constant polynomial.
A polynomial is in which all variables are either in ascending order or descending order is referred to as standard form of polynomial. i.e. the polynomials 8x^{4} + 3x^{3} + 7x^{2} + 3x + 5 and 5 + 3x + 7x^{2} + 3x^{3} + 8x^{4} are in standard form of polynomial. In general, the standard form of polynomial is a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} _ _ _ _ + a_{n}x^{n}, where a_{0}, a_{1}, a_{2}, a_{3} _ _ _ _ _ _ _ _ _ _, a_{n} are all real numbers and n is any whole number.
The coefficient of the highest power of term is called its leading coefficient. Example of leading coefficient is 8x^{4} + 6x^{3} + 8x^{2} - 4x + 2 and its leading coefficient is 8 as the highest coefficient power of x is 4.
In polynomial, the term with highet exponent is called leading term. Example of leading term is 8x^{4} + 3x^{3} + 2x^{2} - 7x + 6. Here, 8x^{4} is the leading term.
Solution
Solution