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# Polynomials: Definitions and Operations

October 24, 2023
written by Rida Mirza

In algebra, polynomials are fundamental algebraic expressions that support a vast array of mathematical concepts and applications.

In this article, we will discuss the world of polynomials, exploring their structure, operations, and practical applications.

Table of Contents

## What Are Polynomials?

Polynomial is an expression consisting of variables and coefficients. It involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

For example, x, 5x^2 – 3x + 7, and 2xy^3 + 4yz are all polynomials.

## General Form of a Polynomial

The general form of a polynomial is:

``f(x) = an * x^n + an-1 * x^(n-1) + ... + a2 * x^2 + a1 * x + a0``

Here, `f(x)` represents the polynomial, `x` is the variable, and `an, an-1, ..., a0` are the coefficients. The exponents `n` should be non-negative integers.

## Degree of a Polynomial

Degree of a polynomial is determined by the term with the highest exponent.

For example, the polynomial 5x^3 + 2x^2 – x + 7 has a degree of 3, since the highest exponent is on the x^3 term.

Degree gives information about the highest power that variable has in the polynomial. A constant polynomial, like 5, has a degree of 0.

## Polynomial vs. Non-Polynomial

Polynomials are distinct from non-polynomial expressions due to their structure. For an expression to be considered a polynomial,

• Variables must have non-negative integer exponents.
• Coefficients should be real numbers.
• There must be a finite number of terms.
• Only addition, subtraction, and multiplication operations are allowed.

## Polynomial Types

### Constant Polynomials (Degree 0)

These have no variable terms. For example, `P(x) = 7`.

### Linear Polynomials (Degree 1)

These have a single variable term. For example, `Q(x) = 3x + 2`.

### Quadratic Polynomials (Degree 2)

These have two variable terms. For example, `R(x) = 2x^2 - 5x + 1`.

### Cubic Polynomials (Degree 3)

These have three variable terms. For example, `S(x) = 4x^3 + x^2 - 3x - 7`.

### Quartic Polynomials (Degree 4)

These have four variable terms. For example, `T(x) = 4x^4 + 3x^3 - x^2 + x - 7`

## Polynomial Operations

### Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms. Like terms have the same variables raised to the same powers.

Example

(3x^2 – 2x + 5) + (2x^2 + 4x – 1) = 5x^2 + 2x + 4

To subtract polynomials, change the sign of the polynomial being subtracted and then combine like terms.

Example

(3x^2 + 5x – 7) – (2x^2 – 4x + 1) = x^2 + 9x – 8

### Multiplying Polynomials

To multiply two polynomials together, use the distributive property and FOIL method. Multiply each term in one polynomial by each term in the other.

Example

(x + 3)(x – 5)

= x(x) – x(5) + 3(x) – 3(5)

= x^2 – 5x + 3x – 15

= x^2 – 2x – 15

The degree of the product is the sum of the degrees of the polynomials being multiplied.

### Dividing Polynomials

Long division can be used to divide polynomials. Divide the first term of the dividend by the first term of the divisor.

Example

(x^3 + 3x^2 – 9) / (x – 3)

The result is x^2 + 6x + 9.

## Special Polynomials

### Monomials

These are polynomials with a single term. For example, `5x` or `4y^3`.

### Binomials

These are polynomials with two terms. For example, `3x - 2` or `a^2 + b`.

## FAQS

### What are the parts of a polynomial called?

The parts of a polynomial are the coefficients and variables. The coefficients are the numbers that are multiplied by the variables. The variables are the unknowns represented by letters like x, y, z.

### What is the degree of a constant polynomial?

A constant polynomial, like 5 or -2, has a degree of 0. This is because it does not contain any variables.

### Can a polynomial have negative exponents?

No, polynomials can only have positive integer exponents. Negative and fractional exponents are not allowed.

### What happens if you divide a polynomial by 0?

Dividing a polynomial by 0 is undefined. It results in an error as you cannot divide any number by 0.

### Do polynomials have limits to how many terms they can have?

No, there is no limit to the number of terms a polynomial can have. However the degree of a polynomial is determined by the greatest exponent on its variable term.

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