Polynomials are algebraic expressions that contain indeterminates and constants. You can think of polynomials as a dialect of mathematics. They are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math, such as calculus. For example, 2x + 9 and x^{2} + 3x + 11 are polynomials. You might have noticed that none of these examples contain the “=” sign. Have a look at this article in order to understand polynomials in a better way.

## What is a Polynomial?

A polynomial is a type of expression. An expression is a mathematical statement without an equal-to sign (=). Let us understand the meaning and examples of polynomials as explained below.

### Polynomial Definition

A **polynomial** is a type of algebraic expression in which the exponents of all variables should be a whole number. The exponents of the variables in any polynomial have to be a non-negative integer. A polynomial comprises constants and variables, but we cannot perform division operations by a variable in polynomials.

### Polynomial Examples

Let us understand this by taking an example: 3x^{2} + 5. In the given polynomial, there are certain terms that we need to understand. Here, x is known as the variable. 3 which is multiplied to x^{2} has a special name. We denote it by the term “coefficient”. 5 is known as the constant. The power of the variable x is 2.

Below given are a few expressions that are not examples of a polynomial.

Not a Polynomial | Reason |
---|---|

2x^{-2} | Here, the exponent of variable ‘x’ is -2. |

1/(y + 2) | This is not an example of a polynomial since division operation in a polynomial cannot be performed by a variable. |

√(2x) | The exponent cannot be a fraction (here, 1/2) for a polynomial. |

The following image shows all the terms in a polynomial.

## Standard Form of Polynomials

The standard form of a polynomial refers to writing a polynomial in the descending power of the variable.

**Example:** Express the polynomial 5 + 2x + x^{2} in the standard form.

To express the above polynomial in standard form, we will first check the degree of the polynomial.

- In the given polynomial, the degree is 2. Write the term containing the degree of the polynomial.
- Now, we will check if there is a term with the exponent of variable less than 2, i.e., 1, and note it down next.
- Finally, write the term with the exponent of the variable as 0, which is the constant term.

Therefore, 5 + 2x + x^{2} in standard form can be written as x^{2 }+ 2x + 5.

Always remember that in the standard form of a polynomial, the terms are written in decreasing order of the power of the variable, here, x.

## Terms of a Polynomial

The terms of polynomials are defined as the parts of the expression that are separated by the operators “+” or “-“. For example, the polynomial expression 2x^{3} – 4x^{2} + 7x – 4 consists of four terms.

### Like Terms and Unlike Terms

Like terms in polynomials are those terms which have the same variable and same power. Terms that have different variables and/or different powers are known as unlike terms. Hence, if a polynomial has two variables, then all the same powers of any ONE variable will be known as like terms. Let us understand these two with the help of examples given below.

**For example**, 2x and 3x are like terms. Whereas, 3y^{4} and 2x^{3} are unlike terms.

## Degree of a Polynomial

The highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. The degree is used to determine the maximum number of solutions of a polynomial equation (using Descartes’ Rule of Signs).

**Example 1:** A polynomial 3x^{4 }+ 7 has a degree equal to four.

The degree of the polynomial with more than one variable is equal to the sum of the exponents of the variables in it.

**Example 2:** Find the degree of the polynomial 3xy.

In the above polynomial, the power of each variable x and y is 1. To calculate the degree in a polynomial with more than one variable, add the powers of all the variables in a term. So, we will get the degree of the given polynomial (3xy) as 2.

Similarly, we can find the degree of the polynomial 2x^{2}y^{4 }+ 7x^{2}y by finding the degree of each term. The highest degree would be the degree of the polynomial. For the given example, the degree of the polynomial is 6.

## Types of Polynomials

Polynomials can be categorized based on their degree and their power. Based on the numbers of terms, there are mainly three types of polynomials that are listed below:

- Monomials
- Binomials
- Trinomials

**Monomial** is a type of polynomial with a single term. For example, x, -5xy, and 6y^{2}. A** binomial** is a type of polynomial that has two terms. For example, x + 5, y^{2 }+ 5, and 3x^{3 }– 7. While a **Trinomial** is a type of polynomial that has three terms. For example 3x^{3 }+ 8x – 5, x + y + z, and 3x + y – 5. However, based on the degree of the polynomial, polynomials can be classified into 4 major types:

- Zero Polynomial
- Constant polynomial
- Linear polynomial
- Quadratic polynomial
- Cubic polynomial

A constant polynomial is defined as the polynomial whose degree is equal to zero. Any constant polynomial with coefficients equal to zero is defined as a **zero polynomial**. For example, 3, 5, or 8. Polynomials with 1 as the degree of the polynomial are called **linear polynomials**. For example, x + y – 4. Polynomials with 2 as the degree of the polynomial are called **quadratic polynomials**. For example, 2p^{2} – 7. Polynomials with 3 as the degree of the polynomial are called **cubic polynomials**. For example, 6m^{3} – mn + n^{2 }– 4.

## Properties of Polynomials

A polynomial expression has terms connected by the addition or subtraction operators. There are different properties and theorems on polynomials based on the type of polynomial and the operation performed. Some of these are as given below,

**Theorem 1:** If A and B are two given polynomials then,

- deg(A ± B) ≤ max(deg A, deg B), with the equality if deg A ≠ deg B
- deg(A⋅B) = deg A + deg B

**Theorem 2:** Given polynomials A and B ≠ 0, there are unique polynomials Q (quotient) and R (residue) such that,

A = BQ + R and deg R < deg B

**Theorem 3 (Bezout’s Theorem):** Polynomial P(x) is divisible by binomial x − a, if and only if P(a) = 0. This is also known as the factor theorem.

**Theorem 4:** If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.

**Theorem 5:** Polynomial P(x) of degree n > 0 has a unique representation of the form P(x) = k(x – x_{1})(x – x_{2})…(x – x_{n}), where k ≠ 0 and x_{1},…,x_{n} are complex numbers, not necessarily distinct.

Therefore, P(x) has at most deg P = n different zeros.

**Theorem 6:** Polynomial of n-th degree has exactly n complex/real roots along with their multiplicities.

**Theorem 7:** If a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by Q⋅R.

**Theorem 8:** If ß is a complex zero of a real polynomial P(x), then so is ¯¯¯ßß¯ (complex conjugate of ß).

**Theorem 9: **A real polynomial P(x) has a unique factorization (up to the order) of the form,

P(x) = (x – r_{1})…(x – r_{k})(x^{2} – p_{1}x + q_{1})…(x^{2} – p_{l}x + q_{l}),

where r_{i} and p_{j}, q_{j }are real numbers with p_{i}^{2} < 4qi and k + 2l = n.

**Theorem 10 (Remainder Theorem): **The remainder when a polynomial f(x) is divided by (x – a) is f(a).

## Operations on Polynomials

The basic algebraic operations can be performed on polynomials of different types. These four basic operations on polynomials can be given as,

- Addition of polynomials
- Subtraction of polynomials
- Multiplication of polynomials
- Division of polynomials

## Addition of Polynomials

Addition of polynomials is one of the basic operations that we use to increase or decrease the value of polynomials. Whether you wish to add numbers together or you wish to add polynomials, the basic rules remain the same. The only difference is that when you are adding, you align the appropriate place values and carry the operation out. However, when dealing with the addition of polynomials, one needs to pair up like terms and then add them up. Otherwise, all the rules of addition from numbers translate over to polynomials. Have a look at the image given here in order to understand how to add any two polynomials.

## Subtraction of Polynomials

As discussed above, the rules for the subtraction of polynomials are very similar to subtracting two numbers. To subtract a polynomial from another, we just add the additive inverse of the polynomial that is being subtracted to the other polynomial. Another easy way to subtract polynomials is, just change the signs of all the terms of the polynomial to be subtracted and then add the resultant terms to the other polynomial as shown below. We just have to align the given polynomials based on the like terms.

## Multiplication of Polynomials

The multiplication operation on polynomials follows the general properties like commutative property, associative property, distributive property, etc. Applying these properties using the rules of exponents we can solve the multiplication of polynomials. To multiply to polynomials, we just multiply every term of one polynomial with every term of the other polynomial and then add all the results. Here is an example to multiply polynomials.

For example, (2x + 3y)(4x – 5y) = 2x(4x – 5y) + 3y(4x – 5y) = 8x^{2} – 10xy + 12xy – 15y^{2}

⇒ 8x^{2} + 2xy – 15y^{2}

## Division of Polynomials

The division of polynomials is an arithmetic operation where we divide a given polynomial by another polynomial which is generally of a lesser degree in comparison to the degree of the dividend. There are two methods to divide polynomials.

To learn more about each type of division, click on the respective link.

## Factorization of Polynomials

Factorization of polynomials is the process by which we decompose a polynomial expression into the form of the product of its irreducible factors, such that the coefficients of the factors are in the same domain as that of the main polynomial. There are different techniques that can be followed for factoring polynomials, given as,

- Method of Common Factors
- Grouping Method
- Factoring by splitting terms
- Factoring Using Algebraic Identities

Based on the complexity of the given polynomial expression, we can follow any of the above-given methods.

## Polynomial Equations

A polynomial equation is an equation formed with variables, exponents, and coefficients together with operations and an equal sign. The general form of a polynomial equation is P(x) = a_{n} x^{n} + . . + rx + s. Some examples of polynomial equations are x^{2} + 3x + 2 = 0, x^{3} + x + 1 = 0, x + 7 = 0, etc.

## Polynomial Functions

The general expressions containing variables of varying degrees, coefficients, positive exponents, and constants are known as polynomial functions. In other words, a polynomial function is a function whose definition is a polynomial. Here are some example of polynomial functions,

- f(x) = x
^{2}+ 4 - g(x) = -2x
^{3}+ x – 7 - h(x) = 5x
^{4}+ x^{3}+ 2x^{2}

## Solving Polynomials

Solving a polynomial means finding the roots or zeros of the polynomials. We can apply different methods to solve a polynomial depending upon the type of the polynomial, whether it is a linear polynomial, quadratic polynomial, and so on. Let us first understand what is meant by the zero of a polynomial.

### Zeros of Polynomials

The roots or zeros of polynomial are the real values of the variable for which the value of the polynomial would become equal to zero. So, if we say any two real numbers, ‘α’ and ‘ß’ are zeroes of polynomial p(x), then p(α) = 0 and p(ß) = 0. For example, for a polynomial, p(x) = x^{2} – 2x + 1, we observe, p(1) = (1)^{2} – 2(1) + 1 = 0. Therefore, 1 is a zero or root of the given polynomial. This also means that (x – 1) is a factor of p(x).

Now, to find the zero or root of any polynomial, that is, to solve any polynomial, we can apply different methods,

- Factorization
- Graphical Method
- Hit and Trial Method

**Important Notes on Polynomials:**

- Terms in a polynomial can be only separated by the ‘+’ or ‘-‘ sign.
- For any expression to become a polynomial, the power of the variable should be a whole number.
- The addition and subtraction of a polynomial are possible between like terms only.
- All the numbers in the universe are called constant polynomials.