In Maths, Polynomial is a concept which is studied under Algebra branch. Before going to definition of a polynomial and its various types, lets go through once the basic terms those comprise polynomials.

Variables are defined and used in algebra. Its definition states that a symbol which can be used to assign different numerical value is known as variable.

Example

\(x,y,z,p,q,r,s\) etc.

In Math, a symbol having a fixed value is called a constant.

Example

\(8, 5, 9, \pi\) etc.

A combination of constants and variables connected by some or all basic operations \(+\), \(-\), \(\times\), \(\div\) is called an algebraic expression.

Example

\(7+8x\)

So, here we have made an algebraic expression connected by \(+\) operation by combining a constant 7 and a constant 8 with variable \(x\) .

lets see another expression.

\(7+8x-6x^2y-{\frac{4}{9}xy}\)

Various parts of an algebraic expression separated by \(+\) or \(-\) operations are called terms.

Example

Consider the algebraic expression as \(7+8x-6x^2y-{\frac{4}{9}xy}\)

So, the terms in this expression are \(7\), \(8x\), \(6x^2y\), \(-\frac{4}{9}xy\)

An algebraic expression in which variables involved are having non negative integral powers is called a polynomial.

Example

We can learn polynomial with two examples:

**Example 1: **\(x^3+2x^2+5x+7\)

Variables involved in the expression is 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: **\(x^2y+xy^2+7y\)

So, variables involved in the expression are x and y.

The powers of x and y in each term are:

\(x^2y\), 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.

Example

\(3x^4+5x^3+7x^2+8\)

This polynomial is one variable polynomial i.e. x.

The powers of the x variable are 4, 3 and 2

\(\therefore\) the highest power is 4.

Hence, the degree of polynomial is 4.

Polynomial has 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 all the three types of polynomial with their examples.

A polynomial containing one non zero term is called monomial.

Example

\(8x^5\)

In this polynomial, there is only one non zero term i.e. \(8x^5\). Therefore, it is an example of monomial.

\(-7y^3\)

Here, \(-7y^3\) is only one non zero term.Therefore, it is also an example of monomial.

A polynomial containing two non zero terms is called binomial.

Example

\(7x^6-3x^4\)

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 trinomial.

Example

\(5x^4+3x^2-8\)

Here we have three non zero terms i.e. \(5x^4\), \(3x^2\) and \(8\). Therefore, it is an example of trinomial.

A polynomial containing four terms is called quadrinomial polynomial.

Example

\(7x^5-3x^2+9x+5\)

A polynomial containing five terms is called quintrinomial polynomial.

Example

\(y^6+8y^5+9y^4+9y^2+7\)

A polynomial of degree 1 is called 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 \(ax+b\)

Example

\(x^1+8\)

A polynomial of degree 2 is called 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 \(ax^2+bx+c\)

Example

\(5x^2+2x-7\)

A polynomial of degree 3 is called 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 \(ax^3+bx^2+cx+d\)

Example

\(2x^3+3x^2-5x+7\)

A polynomial of degree 4 is called 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 \(ax^4+bx^3+cx^2+dx+e\)

Example

\(2x^4-3x^2+5x^3+7x-4\)

A polynomial of degree 0 is called constant polynomial.

Example

\(4, -7\) etc.

**Why?**

\(\because\) \(4\) can be written as \(4y^0\), where degree of this polynomial is zero.

Also, \(-7\) can be written as \(-7y^0\), where 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.

Example

\(0x^5+0x^3+0x^2+0x\) etc.

Note

Degree of zero polynomial is not defined.

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.

- 1) \(3x^3-2x^2\)
- 2) \(4y^5+7y^2\)
- 3) \(11z^4+7z^2+2\)
- 4) \(5t\)
- 5) \(x^3+7x^2+5\)

**Solution**

1) \(3x^3-2x^2\) is a binomial because there exists two terms which are \(3x^3\) and \(2x^2\).

2) \(4y^5+7y^2\) is a binomial because there exists two terms which are \(4y^5\) and \(7y^2\).

3) \(11z^4+7z^2+2\) is a trinomial because there exists three terms which are \(11z^4\), \(7z^2\) and 2.

4) \(5t\) is a monomial because there exists one term which is \(5t\).

5) \(x^3+7x^2+5\) is a trinomial because there exists three terms which is \(x^3\), \(7x^2\) and 5.

- 1) \(7x^2+x-5\)
- 2) \(9x^3+\frac{1}{7}\)
- 3) \(\sqrt{2}x+1\)
- 4) \(x^2+\frac{1}{x}\)
- 5) \(1-x^3\)

**Solution**

1) \(7x^2+x-5\) is a quadratic polynomial as variable x has the highest power of 2.

2) \(9x^3+\frac{1}{7}\) is a cubic polynomial as variable x has the highest power of 3.

3) \(\sqrt{2}x+1\) is a linear polynomial as variable x has the highest power of 1.

4) \(x^2+\frac{1}{x}\) \(\implies\) \(x^2+x^{-1}\)

Hence, it is not a polynomial because x has power of -1, which is a negative integer.

5) \(1-x^3\) is a cubic polynomial as variable x has the highest power of 3.