# Polynomials: Definitions and Operations

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.

**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 Type | Degree | Example |
---|---|---|

Zero Polynomial | Not Defined | P(x) = 0 |

Constant Polynomial | 0 | Q(x) = 8 |

Linear Polynomial | 1 | R(x) = 2x + 5 |

Quadratic Polynomial | 2 | S(x) = 3x^2 + 2x + 1 |

Cubic Polynomial | 3 | T(x) = 4x^3 + 2x^2 + x + 7 |

Quartic Polynomial | 4 | U(x) = 5x^4 + 4x^3 + 2x^2 + 3x + 1 |

**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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