# Binomials

## Related calculator: Polynomial Calculator

**Binomial** is a sum/difference of TWO monomials.

Both monomials are called **terms.**

**Examples of binomials** are:

- $$${5}+{x}$$$
- $$${2}{{x}}^{{2}}+{{y}}^{{3}}{x}$$$
- $$${10}{x}{{y}}^{{3}}-{z}{{y}}^{{2}}{x}$$$

**Examples of expressions, that are not binomials**:

- $$${x}+\frac{{1}}{{x}}$$$ (second term is not a monomial)
- $$$\frac{{{2}-{y}}}{{x}}$$$ (division is not allowed)
- $$${x}{{y}}^{{2}}+{z}-{2}$$$ (binomial can have only TWO terms)

**Degree of the binomial** is the largest number among degrees of its monomials.

For example, in binomial $$${4}{{x}}^{{2}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}$$$ first term has degree $$${2}+{3}={5}$$$ and second term has degree $$${8}+{7}={15}$$$. The largest of numbers 5 and 15 is 15. Thus, degree of the $$${4}{{x}}^{{2}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}$$$ is 15.

**Exercise 1.** Determine whether the following is a binomial: $$$\frac{{2}}{{5}}{x}-\frac{{3}}{{5}}{y}$$$?

**Answer**: yes.

**Exercise 2.** Determine whether the following is a binomial: $$$\frac{{1}}{{x}}+{2}{y}+{3}$$$?

**Answer**: no.

**Exercise 3.** Determine whether the following is a binomial: $$$\sqrt{{{3}}}{x}\cdot{y}+{5}{z}{y}$$$?

**Answer**: yes.

**Exercise 4.** Find degree of the following expression: $$$-{x}{y}+{5}$$$?

**Answer**: 2.

**Exercise 5.** Find degree of the following expression: $$$-{{p}}^{{3}}{{q}}^{{9}}+{5}{{z}}^{{3}}{{y}}^{{4}}{{p}}^{{{10}}}$$$?

**Answer**: 17.