# 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: ${{x}}^{{3}}{y}+{3}{{y}}^{{2}}-{z}$?

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

Exercise 3. Determine whether the following is a binomial: ${3}{{x}}^{{2}}+{2}{x}+{1}$?
Exercise 4. Find degree of the following polynomial: ${{x}}^{{3}}+{2}{x}{{y}}^{{3}}+{{z}}^{{5}}$?
Exercise 5. Find degree of the following expression: ${{x}}^{{3}}+{2}-{{x}}^{{7}}+{5}{{x}}^{{2}}$?