Intro To Polynomials

• The very first function we will learn about is called the polynomial. A polynomial is specifically designed to connect points in any Euclidean Space. For the remainder of the notes, we will specifically look at polynomials in 2 Dimensional Euclidean Spaces(Cartesian/X-Y Plane) and see how we can create function to fit into as many points as we desire!

Polynomials and Their Degree

Def 1) Polynomial

• A polynomial is a function denoted in the following ways : $f(x)=a_n{x}^n+a_{n-1}{x}^{x-1}+...+a_{1}x+a_0$ $y=a_n{x}^n+a_{n-1}{x}^{x-1}+...+a_{1}x+a_0$ $f(x)={\sum }_{i=0}^n{a}_i{x}^i$
• Don’t fret about the 3rd way of writing the polynomial. It’s the most concise way, but it is there for individuals who have been spoiled by higher levels of mathematics.
• In mathematics, you may see polynomial denoted as $p(x)$ instead of the function $f(x)$. Most Pre-Calculus Textbooks default to $f(x)$ so I will also use this convention.

Def 2) Degree of a polynomial

• The degree of the polynomial refers to the largest exponent associated with a variable. In a setence, when someone says “second-degree polynomial”, it will refer to a polynomial like the following $f(x)=a{x}^2+bx+c$ where $a\ne 0$.
• In general, a polynomial is an “nth” degree polynomial if and only if $a_n\ne 0$ as above
• For the purposes of clarity, the most precise way to describe polynomials at the precalculus level would be as “Single Variable Nth Degree Polynomials”, but to be succinct, we will shorten it to “Nth Degree Polynomials”

Examples

$f(x)=x^2-2x+4$ $g(x)=5^{2}x-x^{2}0$

• The degree of $f(x)$ is 2, the degree of $g(x)$ is 20.
• f(x) is a 2nd degree polynomial, g(x) is a 20th degree polynomial
• Both solutions are the same

• For the curious individuals ,below is the rigorous defintion of a polynomial

Rigorous Definition of a Polynomial

a function $p:G\to G$, where G a set (more specifically a group) such that $p(x)={\sum }_{i=0}^n{a}_i{x}^i$

Def 3 Roots of a Polynomial

• A root of a polynomial (AKA x-intercepts) are all values x, such that when you plug it into the polynomial, f(x) = 0

Example

• Observe the following polynomials :

$f(x)=x^2-1$ $g(x)=x^3-x^2$

• When we know the roots of the function we say that :

“The roots of $f(x)$ are $1$ and $-1$” “The roots of $g(x)$ are $0,1$ and $-1$”

• To see why, lets plug in those values into the equations!

$f(1)=1^2-1=1-1=0$ $g(1)=(-1{)}^3-(-1{)}^2=-1+1=0$

Exercise (Evaluate the other roots to make sure they work!)

Warning

Don’t worry about how/why those numbers are roots to the polynomials. Throughout these notes, we will develop techniques to find them!