By definition, a polynomial function is a function that can be written as shown below:
f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x1 + a0
You could also use p(x) instead of f(x) in order to be more precise.
p(x) = anxn + an-1xn-1 + ... + a2x2 + a1x1 + a0
Notice the following important conditions about a polynomial function
- n is a nonnegative integer or an integer that is either positive or zero
- The coefficients an , an-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
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