## What is a Polynomial?

An expression of the form **p(x) = a _{0 }+ a_{1}x + a_{2}x^{2} +….+a_{n}x^{n}, **where a

_{n}≠ 0 is called a polynomial in x of degree n.

Here a_{0, }a_{1}, a_{2},….., a_{n }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 x^{3 })^{,}but only 0, 1, 2, 3, … etc are allowed

**Examples:**

**(i)** 3x + 5 is a polynomial in x of degree 1.

**(ii)** 4y^{2} -3y +8 is a polynomial in x of degree 2.

**(iii)** 2u^{3} + 4u^{2} -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****−4y**^{2}− (\frac{7}{9})x**4xyz + 3xy**^{2}z − 0.1xz − 200y + 0.5**512v**+^{5}**99w**^{5}**9**

(Yes, “9” is a polynomial because it can be just a constant!)

These are **not** polynomials

**5xy**is not polynomials, because the exponent is “-4” (exponents can only be allowed 0,1,2,…)^{-4}**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 | 5x^{2}+5x+1 |

Cubic Polynomial | 3 | 2u^{3}-3u^{3}+8u+1 |

Quartic Polynomial | 4 | 2y^{4}+3y^{3}-5y^{2}+9y+1 |

**Example: Find the degree of the polynomial 4u ^{4} – 5u^{3}+ 6u^{2} -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