# Monomials

## Related Calculator: Polynomial Calculator

**Monomial** is an algebraic expression, that can have the following 3 "parts":

- Number (it is called
**coefficient of monomial**) - Variables, raised to non-negative integer powers
- Operations of multiplication (they "separate" variables)

$$$\huge{\color{green}{\underbrace{15}_{\text{number (coefficient)}}}\color{blue}{\overbrace{\cdot}_{\text{multiplication}}} \color{red}{\underbrace{x^2}_{\text{variable}}} \color{blue}{\overbrace{\cdot}_{\text{multiplication}}} \color{red}{\underbrace{y^3}_{\text{variable}}}}$$$

Any of the above 3 "parts" can be skipped.

**Examples of monomials**:

- `15` (just number is a monomial, variables and multiplication are skipped)
- `x^2` (it can be thought, that there is no coefficient, but it is there! It is 1.)
- `15x` (valid monomial with one variable and multiplication sign, that is not written)
- `2x^2y^3` (monomial with two variables)

Note, that addition, subtraction and division are not allowed for "separating" variables, only for writing coefficient.

**More examples**:

- `x/2=1/2x` (coefficient is `1/2`)
- `(2+sqrt(2))xy^2` (coefficient involves roots)

Now, let's see **examples of expressions, that are not monomials**:

- `2x+y` (addition is used to "separate" variables)
- `y/x^2` (division of variables is not allowed)
- `2^x` (variable exponent is not allowed)
- `2m^(1/3)n^(-2)` (negative and fractional exponents are not allowed).

**Degree of the monomial** is the sum of exponents of all variables it contains.

Since constant monomial doesn't contain variables, its degree equals 0.

**Example**. Degree of `35` is `0`.

**Example**. Degree of `2x` is `1`.

**Example**. Degree of `-5y^2xz^3` is `2+1+3=6`.

**Monomials are called like terms** if they have the same variables to the same power.

For example, `2color(red)(x^5y^7)` and `-4color(red)(x^5y^7)` are like terms, but `2x^3` and `2xy^2` are not.

**Exercise 1**. Determine, whether the following is a monomial: `2x^2y`?

**Answer**: yes.

**Exercise 2**. Determine, whether the following is a monomial: `2/sqrt(3)xy`?

**Answer**: yes.

**Exercise 3**. Determine, whether the following is a monomial: `2x/y`?

**Answer**: no.

**Exercise 4**. Find degree of the monomial `14x^3`.

**Answer**: 3.

**Exercise 5**. Find degree of the monomial `-9x^3a^11p^7`.

**Answer**: 21.