## Polynomials

## What are Polynomials?

**Polynomials are mathematical expressions consisting of variables, coefficients, and non-negative integer exponents.** They take the form of a sum of terms, where each term is a product of a coefficient (a constant number) and a variable raised to an exponent. For example, 3π₯Β²+2π₯β5Β is a polynomial with three terms: 3π₯Β²Β 2π₯, and $5$. Polynomials are used widely in mathematics, science, and engineering to model various phenomena and solve equations, making them essential tools for understanding and applying mathematical concepts.

## Types of Polynomials

Polynomials can be categorized based on the number of terms they contain and their degree:

Monomial | Binomial | Trinomial |

One Term | Two terms | Three terms |

Example: x, 6y, 34, x/2 | Example: xΒ²+x, xΒ³-4x, y+2 | Example: xΒ²+5x+10Polynomial Examples |

Polynomials can be classified in two ways:

**By the Number of Terms:****Monomial:**A polynomial with a single term, such as*x*, β5π₯π¦, or 6π¦Β².**Binomial:**A polynomial with two terms, such as π₯+5, π¦Β²+5, or 3π₯Β³β7.**Trinomial:**A polynomial with three terms, such as 3π₯Β³+8π₯β5, π₯+π¦+π§, or 3π₯+π¦β5.

**By the Degree:**Polynomials can also be classified based on the highest exponent of the variables present:**Linear:**A polynomial of degree 1, such as 2π₯+3.**Quadratic:**A polynomial of degree 2, such as π₯Β²+4π₯+2.**Cubic:**A polynomial of degree 3, such as xΒ³-2xΒ²+3x-4**Quartic:**A polynomial of degree 4, such as π₯β΄β2π₯Β³+π₯Β²+3π₯β5.

## Standard Form of a Polynomial

P(x) = aβxβΏ + aβββx^{βΏβ»ΒΉ} +aβββx^{βΏβ»Β² }+ β¦β¦β¦β¦β¦β¦. + aβx + aβ

Where aβ, a_{βββ}, a_{βββ}, β¦β¦β¦β¦β¦β¦β¦β¦, aβ, aβ are called coefficients of xβΏ, x^{βΏβ»ΒΉ}, x^{βΏβ»Β²}, β¦.., x and constant term respectively and it should belong to real number (β² R).

### Notation

The polynomial function is denoted by P(x) where x represents the variable. For example,

P(x) = x^{Β²}-5x+11

If the variable is denoted by a, then the function will be P(a)

## Degree of a Polynomial

The degree of a polynomial is the highest exponent of its variables in any term. It indicates the complexity and growth rate of the polynomial function.

Polynomial | Degree | Example |
---|---|---|

Zero Polynomial | Not Defined | 4 |

Constant | 0 | P(x) = 4 |

Linear Polynomial | 1 | P(x) = 2x+1 |

Quadratic Polynomial | 2 | P(x) = 2x^{Β²}+4x+2 |

Cubic Polynomial | 3 | P(x) = 6x^{Β³}+3x^{Β²}+2x+4 |

Quartic Polynomial | 4 | P(x) = 8xβ΄+5x^{Β³}+2x^{Β²}+3x+1 |

## Terms of a Polynomial

A polynomial consists of one or more terms, where each term is a product of a coefficient (a constant number) and a variable raised to an exponent. Here are key points to understand about the terms of a polynomial:

**Single-Term:**Polynomials can have a single term, called a monomial, such as 3π₯Β² or 5π¦.**Multi-Term:**Polynomials can also have multiple terms. For example, a binomial has two terms, such as 4π₯+3, while a trinomial has three terms, such as 2π₯Β²+3π₯+1.**Structure:**Each term consists of a variable raised to an exponent (degree) and multiplied by a coefficient. For example, in the term 5π₯Β³, 5 is the coefficient,*x*is the variable, and 3 is the exponent.

## Properties of Polynomials

A polynomial expression consists of terms connected by addition or subtraction operators. There are various properties and theorems associated with polynomials, depending on their type and the operations performed on them. Here are some key properties:

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

- degβ‘(A Β± B) β€ max(degβ‘ A, deg β‘B), with the equality if degβ‘ A β deg β‘B
- degβ‘(Aβ B) = degβ‘ A + degβ‘ B

**Theorem 2:** Given polynomials A and B β 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 β 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_{β})(x – xβ)…(x – xβ), where k β 0 and xβ,β¦,xβ are complex numbers, not necessarily distinct.

Therefore, P(x) has at most deg β‘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 Qβ
R.

**Theorem 8:** If Γ is a complex zero of a real polynomial P(x), then so is Β―Β―Β―ΓΓΒ― (complex conjugate of Γ).

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

P(x) = (x – rβ)…(x – rβ)(xΒ² – pβx + qβ)…(xΒ² – pβx + qβ),

where rα΅’ and pβ±Ό, q_{β±Ό }are real numbers with p_{i}Β² < 4qi and k + 2l = n.

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

**Theorem 11 (Rational Root Theorem): **A rational root of a polynomial functional f(x) = aβxβΏ + aβββx^{βΏβ»ΒΉ} + … + aβxΒ² + aβx + aβ is of the form p/q, where p is a factor of a_{0} and q is a factor of a_{n}. This theorem is very helpful in finding the rational zeros of a polynomial.

## Operations on Polynomials

Polynomials can be manipulated using four basic algebraic operations, making them versatile tools in mathematics. Here’s a detailed look at each operation:

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

**Addition of Polynomials:**Polynomials can be added by combining like terms, which are terms with the same variable raised to the same exponent. For example, (3π₯Β²+2π₯)+(4π₯Β²+5π₯) simplifies to 7π₯Β²+7π₯ by adding the like terms 3π₯Β² and 4π₯Β², and 2π₯ and 5π₯.**Subtraction of Polynomials:**Subtracting polynomials involves combining like terms in a similar way to addition, but with subtraction. For example, (5π₯Β³β2π₯)β(3π₯Β³+4π₯)simplifies to 2π₯Β³β6π₯ by subtracting the like terms.**Multiplication of Polynomials:**When multiplying polynomials, each term of the first polynomial is multiplied by each term of the second polynomial, and the resulting terms are combined. For instance, (2π₯+3)Γ(π₯β1) simplifies to 2π₯Β²β2π₯+3π₯β3 and further to 2π₯Β²+π₯β3.**Division of Polynomials:**Dividing polynomials can be more complex. The division process, known as polynomial long division or synthetic division, involves dividing the terms of the dividend by the terms of the divisor, similarly to numerical division. For example, dividing (4π₯Β²β5π₯+1) by (π₯β1) results in a quotient (4π₯β1) and a remainder, which can further refine the result.

## Factorization of Polynomials

Factorization of polynomials is the process of breaking down a polynomial into simpler, multiplicative components, or factors, that can be multiplied together to reconstruct the original polynomial. Here are key points to understand about this process:

**Common Factor:**The first step in factorization is often finding a common factor across all terms. For example, in the polynomial 6π₯Β²+9π₯, 3x is a common factor, allowing it to be factored as 3π₯(2π₯+3).**Quadratic Factorization:**Quadratic polynomials of the form ππ₯Β²+ππ₯+π can often be factored into two binomial terms. For instance, π₯Β²+5π₯+6 factors to (π₯+2)(π₯+3).**Difference of Squares:**Polynomials of the form πΒ²βπΒ² can be factored using the difference of squares formula (πβπ)(π+π). For example, π₯Β²β16 factors to (π₯β4)(π₯+4).**Higher-Degree Polynomials:**Factoring higher-degree polynomials can be more complex and may involve techniques such as synthetic division or the rational root theorem to find potential roots, which then lead to factorable forms.**Applications:**Factorization simplifies polynomial expressions, making them easier to solve or analyze, particularly in algebraic equations. It also helps in finding the roots of polynomials and in simplifying rational expressions.

## How to Solve

**Example 1: Addition of Polynomials:** Add the polynomials 3π₯Β²+4π₯β5 and 2π₯Β²β3π₯+6

**Solution:** Combine like terms:

(3*x*Β²+4*x*β5)+(2*x*Β²β3*x*+6)=3*x*Β²+2*x*Β²+4*x*β3*x*β5+6

= 5*x*Β²+*x*+1

The resulting polynomial is 5π₯Β²+π₯+1.

**Example 2: Subtraction of Polynomials:** Subtract the polynomial 4π₯Β²β3π₯+2 from 6π₯Β²+π₯β5

**Solution:** Combine like terms:

(6*x*Β²+*x*β5)β(4*x*Β²β3*x*+2)=6*x*Β²β4*x*Β²+*x*+3*x*β5β2

=2*x*Β²+4*x*β7

The resulting polynomial is 2*x*Β²+4*x*β7

**Example 3: Multiplication of Polynomials:** Multiply the polynomials 2π₯+3 and π₯β1

**Solution:** Distribute each term of the first polynomial by each term of the second:

(2x+3) X (xβ1)=2 X Γ(xβ1)+3Γ(xβ1)

= 2xΒ²β2x+3xβ3

=2xΒ²+xβ3

**Example 4: Division of Polynomials:** Divide the polynomial 4π₯Β²β8π₯+4 by 2π₯β2

**Solution:** Using polynomial division:

(4*x*Β²β8*x*+4)/(2*x*β2)β=(2*x*β2)Β²/(2*x*β2)=2*x*β2

The quotient is 2π₯β2.

## Faqs

What is a real life example of a polynomial?

A real-life example of a polynomial is calculating the area of a rectangular field. The area π΄=πΓπ€ can be expressed as a polynomial, for example π΄=5π₯+3 if length π=5π₯and width π€=3. Polynomials are also used in financial modeling and physics.

### What is not a polynomial?

An expression is not a polynomial if it contains variables raised to negative or fractional exponents, variables inside radicals, or variables in the denominator. For example, 1/π₯β and π₯βΒΉ are not polynomials because they don’t meet these criteria.

### How do you identify a polynomial?

A polynomial consists of terms with variables raised to non-negative integer exponents and multiplied by constants. Like terms can be combined through addition or subtraction. The polynomial 4π₯Β²β3π₯+2 is an example of a valid polynomial.

### Is 2xβ΅ a polynomial?

Yes, 2xβ΅ is a polynomial. It consists of a single term where a variable π₯ is raised to a non-negative integer exponent (5), and it is multiplied by a constant coefficient (2). This satisfies the definition of a polynomial.