## What is a Polynomial?

A polynomial is a mathematical equation that consists of variables and coefficients. It involves only the operations of addition, subtraction, and multiplication. The variables in a polynomial can have non-negative integer exponents.

For example, the expression 3x^{2} – 2x + 5 is a polynomial.

where,

- 3, -2, and 5 are coefficients,
- x is a variable,
- and 2 and 1 are the exponents.

The highest exponent in this polynomial is 2, so it is called a **quadratic polynomial**. Polynomials can have any degree, which is the highest exponent of the variable in the polynomial.

**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 the highest power (or exponent) of the variable in the polynomial equation.

For example, in the polynomial expression **4u ^{4} – 5u^{3}+ 6u^{2} -8u+3**, the degree of the polynomial is 4, which is the highest power of the variable u. This is a fourth-degree polynomial.

Here are given below some properties described to the degree of a polynomial:

- The degree of a non-zero constant polynomial is 0.
- The degree of a non-zero linear polynomial in one variable is 1.
- The degree of the sum or difference of two polynomials is the highest degree of the two polynomials.
- The degree of the product of two polynomials is the sum of the degrees of the two polynomials.

Here are some examples of polynomials of different degrees:

- Degree 0: Constant polynomial (e.g. 6)
- Degree 1: Linear polynomial (e.g. 5x – 6)
- Degree 2: Quadratic polynomial (e.g.
**x**+ 2x + 1)^{2} - Degree 3: Cubic polynomial (e.g. 2
**x**– 5^{3}**x**+ 3x + 7)^{2} - Degree n: nth degree polynomial (e.g. 4
**x**+ 2^{n}**x**+ 5^{n-1}**x**} – 3)^{n-2}

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 |

### Related Topics

**Example: Find the degree of the polynomial 4u ^{4} – 5u^{3}+ 6u^{2} -8u+3**

**Solution:**

The degree of the polynomial is 4.

## Operations on Polynomials

Several operations can be performed on polynomials. These operations are the basis for solving polynomials.

Here are given some of the most common operations on polynomials:

**1. Addition and Subtraction**

Polynomials can be added and subtracted by combining like terms.

**For example,** (3x^{2} + 2x + 1) + (2x^{2} – 3x + 2) = 5x^{2} – x + 3.

**2. Multiplication**

Polynomials can be multiplied using the distributive property.

** For example,** (x + 2)(x – 3) = x^{2} – x – 6.

**3. Division**

Polynomials can be divided using long division or synthetic division.

**For example,** (x^{2} + 2x + 1) ÷ (x + 1) = x + 1.

**4. Evaluation**

Polynomials can be evaluated for a specific value of the variable.

** For example,** P(x) = x^{2} + 2x + 1, P(2) = 9.

**5.**** Factoring**

Polynomials can be factored into their constituent factors.

**For example,** x^{2} + 3x + 2 can be factored as (x + 1)(x + 2).

**6. Integration**

The integral of a polynomial can be found using the power rule.

** For example**, if P(x) = x^{3} + 2x^{2} + x, then the indefinite integral of P(x) with respect to x is (1/4)x^{4} + (2/3)x^{3} + (1/2)x^{2} + C, where C is a constant.

**7. Derivatives**:

The derivative of a polynomial can be found using the power rule.

** For example,** if P(x) = x^{3} + 2x^{2} + x, then P'(x) = 3x^{2} + 4x + 1.

## 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