# Polynomials Polynomials are [[Function (in Algebra)|functions]] that are made by combining constants and variables using addition and multiplication. Generally any polynomial can be written as $ f(x) = \sum_{k=0}^n a_kx^k $ Where $k$ is the polynomial's power and $a$ are coefficients of each individual term. **All $k$ powers must be integers or a function is not a polynomial.** All polynomials are smooth functions. That is to say their domain includes all real numbers, and all of its derivatives are continuous (a function is continuous if for all $a$ elements of the domain the following holds $\lim_{x \to a} f(x) = f(a)$). For proof see [[Limit#The Polynomial Rule]]. First degree polynomials are [[Linear Equations]], while second degree polynomials are called [[Quadratics]]. ^gfhhvm See [[Descartes' Rule of Signs]] for a theorem linking coefficients to the number of a polynomial's real roots. For transformations, see [[Function Transformations]]. Polynomials can be divided using long division, or more simply using a technique called synthetic division (https://www.purplemath.com/modules/synthdiv.htm). ## Polynomial Root A polynomial's root is an $x$ value for which the polynomial is 0. Put another way, it's where its graph intersects (or touches) the $x$ axis. Each root has [[Polynomials#Root Multiplicity]] whose parity determines if the $x$ axis is touched or intersected. [[The Fundamental Theorem of Algebra]] states that any polynomial with real coefficients has $n$ roots, where $n$ is the polynomial's power. The [[Rational Root Theorem]] defines the form of any real roots of polynomials with integer coefficients. [[A Complex Conjugate of a Root of a Polynomial with Real Coefficients Is Also a Root]] and from this follows that [[All Polynomials With Real Coefficients Can Be Factored Into Linear And Irreducible Quadratic Terms]] and thus that [[All Odd Degree Polynomials With Real Coefficients Must Have At Least One Real Root]]. ### Root Multiplicity Root multiplicity of root $r$ of [[Polynomials]] $f(x)$ is the highest integer $k$ such that $(x-r)^k$ divides $f(x)$. **The root multiplicity tells us how a polynomial's graph will behave around the root. If the multiplicity is even, the graph will only touch the $x$-axis, if it is odd it will cross the $x$-axis.** You can reason with this by plotting the graphs, and also by analyzing the intervals of the root. Take $f(x) = (x-1)(x-4)^2$ as an example. If $x<1$, the sign of the first expression will be negative and the sign of the second one positive. If however $x>1$, the sign of the first expression will be flipped (because of the odd power), but the sign of the second one will stay the same because the even power will always lead to a positive value. ## Polynomial End Behavior A polynomial's end behavior dictates how it behaves as $x$ approaches positive and negative infinity. It can either be up/up, down/down, up/down or down/up, depending if the $y$ values approach positive (up) or negative (down) infinity. **If the end behavior is the same (up/up or down/down) the polynomial is said to be an *even polynomial*, and if the end behavior is different then the polynomial is said to be an *odd polynomial*. The naming comes from the parity of the polynomial's degree.** If the degree is even, the sign of its output will always be the same and will match the sign of the leading coefficient (provided the constant term is 0). **The consequence of this is that any polynomial of an odd degree with real coefficients will have at least one real root, as it will have to cross the $x$ axis at least once.** For another way of showing this to be the case see [[All Odd Degree Polynomials With Real Coefficients Must Have At Least One Real Root]]. ## Graph Symmetry ![[Graph Symmetry#^kbz2ya]] For polynomials, this happens if all of its odd degree coefficients are 0. ![[Graph Symmetry#^7ca4th]] For polynomials, this happens if all of its even degree coefficients are o (including the constant term). ## Monic Polynomial A monic polynomial is a kind of polynomial that has a single-variable[^1] and whose leading coefficient is 1. Dealing with monic polynomials is often simpler, especially when trying to find roots, as they allow for the use of the [[The Quadratic Formula#Simpler Version|the simpler version of the quadratic formula]], and [[Vieta's Formulas]]. Use of the [[Rational Root Theorem]] to guess rational solutions is also simpler. [^1]: Also known as *univariate polynomial*