# Monomials

**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)

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

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.)
- $$${15}{x}$$$ (valid monomial with one variable and multiplication sign, that is not written)
- $$${2}{{x}}^{{2}}{{y}}^{{3}}$$$ (monomial with two variables)

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

**More examples**:

- $$$\frac{{x}}{{2}}=\frac{{1}}{{2}}{x}$$$ (coefficient is $$$\frac{{1}}{{2}}$$$)
- $$${\left({2}+\sqrt{{{2}}}\right)}{x}{{y}}^{{2}}$$$ (coefficient involves roots)

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

- $$${2}{x}+{y}$$$ (addition is used to "separate" variables)
- $$$\frac{{y}}{{{x}}^{{2}}}$$$ (division of variables is not allowed)
- $$${{2}}^{{x}}$$$ (variable exponent is not allowed)
- $$${2}{{m}}^{{\frac{{1}}{{3}}}}{{n}}^{{-{2}}}$$$ (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 $$${2}{x}$$$ is $$${1}$$$.

**Example.** Degree of $$$-{5}{{y}}^{{2}}{x}{{z}}^{{3}}$$$ is $$${2}+{1}+{3}={6}$$$.

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

For example, $$${2}{\color{red}{{{{x}}^{{5}}{{y}}^{{7}}}}}$$$ and $$$-{4}{\color{red}{{{{x}}^{{5}}{{y}}^{{7}}}}}$$$ are like terms, but $$${2}{{x}}^{{3}}$$$ and $$${2}{x}{{y}}^{{2}}$$$ are not.

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

**Answer**: yes.

**Exercise 2.** Determine, whether the following is a monomial: $$$\frac{{2}}{\sqrt{{{3}}}}{x}{y}$$$?

**Answer**: yes.

**Exercise 3.** Determine, whether the following is a monomial: $$${2}\frac{{x}}{{y}}$$$?

**Answer**: no.

**Exercise 4.** Find degree of the monomial $$${14}{{x}}^{{3}}$$$.

**Answer**: 3.

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

**Answer**: 21.