# What are Polynomials? – Definition, Types, Examples

Polynomials are algebraic expressions with variables and coefficients in them. Polynomials are a type of mathematical dialect. They are used to express numbers in almost every field of mathematics and play an essential role in others, such as calculus.

Polynomials include \(2x+9\), \(x^{2}+3x+11\). You may have observed that none of these examples uses the “=” symbol. Check out this article to have a better understanding of polynomials.

**What is a Polynomial?**

The word polynomial originated from the Greek ‘poly’, which means ‘many’, and ‘nominal’ means ‘terms’ therefore, altogether, it is described as “many terms”.

A polynomial is a mathematical expression that does not contain the equal to sign (=). Let us look at the definition and examples of polynomials given below.

**Definition of Polynomial**

A polynomial is a sort of algebraic statement in which all variable exponents are whole numbers. Variable exponents in any polynomial must be non-negative integers. A polynomial is made up of constants and variables, where we cannot divide polynomials by variables.

**Notation and Standard Form of Polynomial**

**\(p(x)= a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+……+ a_{1}x+ a_{0}\)**

Where, \(a_{n}, a_{n-1}, a_{n-2},….\) are termed coefficients of \(x^{n}, x^{n-1}, x^{n-2},….\) and constant term, correspondingly, and it should be a real number (⋲ R).

**Types of Polynomials**

Polynomials are classified into the following groups based on the number of terms:

- Monomial
- Binomial
- Trinomial

A polynomial with four terms (Quadronomial)

Polynomial with 5 terms (pentanomial), and so on…

These polynomials can be coupled with addition, subtraction, multiplication, and division, but never with a variable.

**Monomial**

A monomial expression has only one term. The single term in an expression must be non-zero for it to be a monomial. These are some examples of monomials:

\(5x\)

\(3\)

\(6 a^{4}\)

\(-3xy\)

**Binomial**

A binomial expression has two terms. A binomial is the sum or difference of two or more monomials. These are some examples of binomials:

\(-5x+3\)

\(6 a^{4}+17x\)

\(x y^{2}+xy\)

**Trinomial**

A trinomial expression has three terms. Below are some examples of trinomial expressions:

\(-8 a^{3}+4x+7\)

\(4 x^{2}+9x+7\)

**Examples of Polynomials**

Let me illustrate this using an example: 3×2 + 5. We need to understand the given terms. In this situation, the variable is x. Where product 3 multiplied by x2 has a different name. The term “coefficient” is used to describe it. The number 5 is known as the constant. The variable x has a power of two.

**Polynomial Examples**

**Polynomial Addition Examples**

**Ex. Add \(x^{2}-x+5\) and \(6 x^{2}+2x-10\)**

**Ans. ** As given, Let’s add the above equations. Then,

Then

\(=>x^{2}-x+5+6 x^{2}+2x-10\)Grouping Like terms,

\((x^{2}+6 x^{2})+(-x+2x)+(5-10)\)**\(=>7 x^{2}+x-5\)**

**Polynomial Subtraction Examples**

**Ex. Subtract \(2 x^{2}-6x+12\) from \(3 x^{2}-8x+7\)**

**Ans. **As given, Let’s Subtract the above equations.

Then

\(3 x^{2}-8x+7-2 x^{2}+6x-12\)Grouping the Like Terms,

\((3 x^{2}-2x^{2})+(-8x+6x)+(7-12)\)**\(x^{2}-2x-5\)**

**Properties of Polynomials**

Several key polynomial characteristics, as well as some notable polynomial theorems, are as follows:

**Property 1:** Division Algorithm

When a polynomial P(x) is divided by a polynomial G(x), the quotient Q(x) with the remainder R(x) is obtained.

P(x) = G(x) • Q(x) + R(x)

Where R(x)=0 or the degree of R(x) < the degree of G(x)

**Property 2:** Bezout’s Theorem

Polynomial P(x) is divisible by binomial (x – a) if and only if P(a) = 0.

**Property 3:** Remainder Theorem

If P(x) is divided by (x – a) with remainder r, then P(a) = r.

**Property 4:** Factor Theorem

A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).

**Property 5:** Intermediate Value Theorem

If P(x) is a polynomial, and P(x) ≠ P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].

**Property 6**

The addition, subtraction, and multiplication of polynomials P and Q are

Degree (P ± Q) ≤ Degree (P or Q)

Degree (P × Q) = Degree(P) + Degree(Q)

**Property 7**

Every zero of Q is likewise a zero of P if a polynomial P is divisible by a polynomial Q.

**Property 8**

P is divisible by (Q • R) if it is divisible by two co-prime polynomials Q and R.

**Property 9**

If, \(p(x)= a_{0}+ a_{1}x+ a_{2} x^{2}+…..+ a_{n} x^{n}\) is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.

**Property 10:** Descartes’ Sign Rule

The number of positive real zeroes in a polynomial function P(x) is equal to or less than the number of changes in the sign of the coefficients by an even integer. If the “K” sign changes, the number of roots is “k” or “(k – a)”, where “a” is an even integer.

**Property 11: **Fundamental Theorem of Algebra

There is at least one complex zero in every non-constant single-variable polynomial with complex coefficients.

**Property 12**

f P(x) is a real-coefficient polynomial with one complex zero (x = a – bi), then x = a + bi is also a zero of P. (x).

Therefore, \(x^{2}-2ax+ a^{2}+ b^{2}\) will be a P factor (x).

**Polynomial Operations**

Basic algebraic procedures can be applied to polynomials of various forms. These four fundamental polynomial operations are as follows:

- Addition of polynomials
- Subtraction of Polynomial
- Polynomial multiplication
- Dividing polynomials

**Addition of Polynomials**

Adding polynomials always requires similar terms that are with the same variable and power. In the Polynomial addition, the result is the same polynomial degree.

**Subtraction of Polynomials**

Subtracting polynomials is the same as addition, the only difference is the style of operation. Where you have to subtract the like terms. Also, note that the subtraction of polynomials results in the same degree polynomial.

**Multiplication of Polynomials**

When two or more polynomials are multiplied, they always result in a higher degree polynomial.

**Division of Polynomials**

A polynomial can be formed by dividing two polynomials. Let’s take a closer look at polynomial division. Follow these steps to divide polynomials:

**Polynomial Division Steps**

- We utilize the long division approach when a polynomial contains more than one term. The steps are as follows.
- In decreasing sequence, write the polynomial.
- Examine the greatest power and divide the phrases by it.
- As the division symbol, use the solution from step 2.
- Subtract it from the following word and bring it down.
- Steps 2–4 should be repeated until there are no more words to carry down.
- It is important to note that the final answer, including the remainder, will be in fraction form (last subtract term).

**Examples of Polynomial Operations**

**Addition of Polynomials**

**Ex. \((2 x^{2}+3x+2)+(3 x^{2}-5x-1)\)**

**Ans. **\(2 x^{2}+3x+2+3 x^{2}-5x-1\)

Adding like terms,

\((2 x^{2}+3 x^{2})+(3x-5x)+(2-1)\)Finally,

\(5 x^{2}-2x+1\)**Multiplication of Polynomials**

**Ex. \((6x−3y) × (2x+5y)\)**

**Ans.** As Given,

Therefore,

\((12 x^{2}+30xy)-(6yx+15 y^{2})\)Then,

\(12 x^{2}+24xy-15 y^{2} \)**Final Notes**

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