# Polynomials

## Related Calculator: Polynomial Calculator

**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`
- `2x^2+y^3x+zy^3`
- `10xy^3-zy^2x`

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

- `x+1/x+3` (second term is not a monomial)
- `(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 `4x^4y^3-9z^8y^7+3x^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 `4x^4y^3-9z^8y^7+3x^2` is 15. Its leading term is `-9z^8y^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_nx^n+a_(n-1)x^(n-1)+...+a_2x^2+a_1x+a_0`, where `n` is positive integer.

Using above definitions, we find, that degree of such polynomial is `n`, leading term is `a_nx^n` and leading coefficient is `a_n`.

**Examples of polynomials in one variable**:

- `-2y^5+y^4+3y^2` (degree is 5, leading term is `-2y^5`, leading coefficient is `-2`)
- `1+x^3+x^2-2x^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, `2x+3`. - 2nd degree:
**quadratic**. For example, `x^2-2x+5`. - 3rd degree:
**cubic**. For example, `-4x^3+2x^2-5`. - 4th degree:
**quartic**. For example, `3x^4-2x^3+x^2+x+7`.

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

**Answer**: yes.

**Exercise 2**. Determine whether the following is a binomial: `sqrt(y)+x^2y`?

**Answer**: no.

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

**Answer**: yes.

**Exercise 4**. Find degree of the following polynomial: `x^3+2xy^3+z^5`?

**Answer**: 5.

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

**Answer**: 7.