# Polynomial functions

### Polynomial functions

#### Lessons

A polynomial function is a function in the form:

$f\left( x \right)\; = {a_n}{x^n} + \;{a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} +$$+ {a_2}{x^2} + {a_1}x + {a_0}$

$\bullet$coefficients: ${a_n}$, ${a_{n - 1}}$, . . . , ${a_2}$, ${a_1}$
$\bullet$leading coefficient: "${a_n}$", the coefficient of the highest power of x
$\bullet$constant term: "${a_0}$", the term without $x$
$\bullet$degree of the polynomial function: $n$, the highest power of $x$
• Introduction
Introduction to Polynomial Functions
$\cdot$ What is a polynomial function?
$\cdot$ Exercise:
State the degree, leading coefficient and constant term for the following polynomial functions:
1) $f(x) = 9x^{5}+7x^{4}-2x^{3}-12x^{2}+x-10$
2) $p(x) = -23x^{18}+37x^{15}-11x^{58}+6$

• 1.
Recognizing a Polynomial Function

Which of the following are not polynomial functions? Explain.

a)
$f(x) = 5x^{2}+4x-3x^{-1}+2$

b)
$f(x) = -x^{3}+6x^{\frac{1}{2}}$

c)
$f(x) = (\sqrt x + 3)(\sqrt x - 3)$

d)
$f(x) = x^{5}+\pi x-\sqrt7 x^{2}+\frac{3}{11}$

• 2.
Classifying Polynomial Functions by Degree

Complete the chart: • 3.
Classifying Polynomial Functions by Number of Terms
Write a polynomial satisfying the given conditions:
i) monomial and cubic
ii) binomial and linear
iii) trinomial and quartic