Polynomial function

By definition, a polynomial function is a function that can be written as shown below:

f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x¹ + a₀

You could also use p(x) instead of f(x) in order to be more precise.

p(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x¹ + a₀

Notice the following important conditions about a polynomial function

The coefficients a_n , a_n-1 , ... , a₂ + a₁ , a₀ are real numbers and they are constants

The terms in the polynomial function above are in descending order by degree

A polynomial is in standard form if the terms in the polynomial are in descending order by degree.

Notice that the exponents n, n-1, ... 2, 1 are whole numbers. If the exponent is negative, then it is not a polynomial.

For example, the following is a polynomial function.

p(x) = -2x⁵ + 6x⁴ + 10x³ + -3x² + 5x + 9

-2x⁵ is the quintic term

6x⁴ is the quartic term

10x³ is the cubic term

-3x² is the quadratic term

5x is the linear term

9 is the constant term

The degree of a polynomial function is the biggest degree of any term of the polynomial. Therefore, the degree of the polynomial function above is 5.

The leading term is the term with the biggest degree. Therefore, the leading term of the polynomial function above is the quintic term or -2x⁵

The leading coefficient is the coefficient of the leading term. The leading coefficient is -2 then for the polynomial function above.

How to name a polynomial function

p(x) = 6x⁴ + 5x is a quartic binomial

p(x) = -2x⁵ + 6x⁴ + 9 is a quintic trinomial

p(x) = -2x⁵ + 6x⁴ + 10x³ + -3x² + 5x + 9 is a quintic polynomial of 5 terms

p(x) = 5x + 9 is a linear binomial

p(x) = -3x² is a quadratic monomial

p(x) = -10x³ + -3x² + 5x + 9 is a cubic polynomial of 4 terms