# Polynomial functions

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### Introduction

#### Lessons

1. 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) $p(x) = -23x^{18}+37x^{15}-11x^{58}+6$
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### Examples

#### Lessons

1. Recognizing a Polynomial Function

Which of the following are not polynomial functions? Explain.

1. $f(x) = 5x^{2}+4x-3x^{-1}+2$
2. $f(x) = -x^{3}+6x^{\frac{1}{2}}$
3. $f(x) = (\sqrt x + 3)(\sqrt x - 3)$
4. $f(x) = x^{5}+\pi x-\sqrt7 x^{2}+\frac{3}{11}$
2. Classifying Polynomial Functions by Degree

Complete the chart: 1. 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

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###### Topic Basics

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 $x$