# Polynomials

**Polynomial** is a monomial or sum/difference of monomials.

Monomials are called **terms** of the polynomial.

Note, that binomial is also polynomial.

**Examples of polynomial** are:

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

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

- $$${x}+\frac{{1}}{{x}}+{3}$$$ (second term is not a monomial)
- $$$\frac{{{2}-{x}}}{{{x}+{2}}}$$$ (division is not allowed)

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

**Leading term** is a monomial with the largest degree. **Leading coeffcient** is a coefficient of the leading term.

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

Thus, degree of the $$${4}{{x}}^{{4}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}+{3}{{x}}^{{2}}$$$ is 15. Its leading term is $$$-{9}{{z}}^{{8}}{{y}}^{{7}}$$$ and leading coefficient is $$$-{9}$$$.

**Polynomial in one variable** is a polynomial, that contains only one variable.

In general, it can be written as $$${a}_{{n}}{{x}}^{{n}}+{a}_{{{n}-{1}}}{{x}}^{{{n}-{1}}}+\ldots+{a}_{{2}}{{x}}^{{2}}+{a}_{{1}}{x}+{a}_{{0}}$$$, where $$${n}$$$ is positive integer.

Using above definitions, we find, that degree of such polynomial is $$${n}$$$, leading term is $$${a}_{{n}}{{x}}^{{n}}$$$ and leading coefficient is $$${a}_{{n}}$$$.

**Examples of polynomials in one variable**:

- $$$-{2}{{y}}^{{5}}+{{y}}^{{4}}+{3}{{y}}^{{2}}$$$ (degree is 5, leading term is $$$-{2}{{y}}^{{5}}$$$, leading coefficient is $$$-{2}$$$)
- $$${1}+{{x}}^{{3}}+{{x}}^{{2}}-{2}{{x}}^{{2}}$$$ (degree is 3, leading term is $$${{x}}^{{3}}$$$, leading coefficient is $$${1}$$$)

Depending on the degree, polynomial in one variable has different names:

- zero degree:
**constant.**For example, $$${7}$$$. - 1st degree:
**linear.**For example, $$${2}{x}+{3}$$$. - 2nd degree:
**quadratic.**For example, $$${{x}}^{{2}}-{2}{x}+{5}$$$. - 3rd degree:
**cubic.**For example, $$$-{4}{{x}}^{{3}}+{2}{{x}}^{{2}}-{5}$$$. - 4th degree:
**quartic.**For example, $$${3}{{x}}^{{4}}-{2}{{x}}^{{3}}+{{x}}^{{2}}+{x}+{7}$$$.

**Exercise 1.** Determine whether the following is a polynomial: $$${{x}}^{{3}}{y}+{3}{{y}}^{{2}}-{z}$$$?

**Answer**: yes.

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

**Answer**: no.

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

**Answer**: yes.

**Exercise 4.** Find degree of the following polynomial: $$${{x}}^{{3}}+{2}{x}{{y}}^{{3}}+{{z}}^{{5}}$$$?

**Answer**: 5.

**Exercise 5.** Find degree of the following expression: $$${{x}}^{{3}}+{2}-{{x}}^{{7}}+{5}{{x}}^{{2}}$$$?

**Answer**: 7.