# 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?

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

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