Polynomial

Definitions

• A polynomial function of degree $n$ is a function of the form $p(x)=\sum _{i=0}^n{a}_i{x}^i$ where $n$ is a nonnegative integer and and $a_n\ne 0$,
• $a_0,a_1,\dots ,a_n$ are called the coefficients of the polynomial
  • $a_0$ is the constant coefficient or constant term
  • $a_n$ is the leading coefficient, and $a_n{x}^n$ is the leading term
• $n=\mathrm{deg}p$ is the polynomial degree
• A number $c$ is called a root of $p(x)$ if $p(c)=0$
• $p(x)=0$ is the zero polynomial. (its degree is $-1$, $-\infty$ or undefined)
• $p_2\mid p_1$ ($p_2$ divides $p_1$, or $p_1$ is divided by $p_2$) if there exists polynomial $q$ such that $p_1=p_{2}q$
• A monic polynomial is a non-zero polynomial in which the leading coefficient is equal to 1.
  • $x^n+c_{n-1}{x}^{n-1}+\cdots +c_2{x}^2+c_{1}x+c_0,$
• A trinomial is a polynomial consisting of three terms or monomials

Theorems

• Fundamental Theorem of Algebra (Equivalent statements)

  • Every univariate polynomial of positive degree with real coefficients has at least one complex root.
  • every non-zero, single-variable, degree $n$ polynomial with complex coefficients has, counted with multiplicity, exactly $n$ complex roots
  • Every univariate polynomial of positive degree $n$ with complex coefficients can be factorized as $c(x-r_1)\cdots (x-r_n)$, where $c,r1,\dots ,r_n$ are complex numbers.

• every non-zero, degree $n$ polynomial has at most $n$ real roots

• every polynomial is a continuous function on $\mathbb{R}$

• Factor theorem: $p(c)=0⟺p(x)=(x-c)q(x)$. (where $\mathrm{deg}q=(n-1)$

  • In words: $c$ is a root of the polynomial $p$ (of degree n) if and only if $(x-c)$ is a factor of $p$

• Let $p:\mathbb{R}\to \mathbb{R}$ be a polynomial of odd degree

  • $\exists c\in \mathbb{R}:p(c)=0$ (in words, $p$ has at least one real root)
  • $p$ is surjective

• Remainder Theorem - If a polynomial $p(x)$ is divided by $x-k$, then the remainder is $r=p(k)$.

• Factors of a Polynomial - Every polynomial of degree $n>0$ with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.todo

• Descartes’ rule of signs - The number of strictly positive roots (counting multiplicity) of $p$ is equal to the number of sign changes in the coefficients of $p$, minus a nonnegative even number

  • The number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by $-1$, or fewer than it by an even number

• $\mathrm{deg}pq=\mathrm{deg}p+\mathrm{deg}q$ (where $p,q$ are not zero polynomial)

• $\mathrm{deg}(p+q)\le \max \{\mathrm{deg}p,\mathrm{deg}q\}$ (where $p,q$ are not zero polynomial)

• If $p$ has $n$ distinct roots, then $p^{\prime }$ has at least $n-1$ roots

• if even degree polynomial $p$ has a root $c$ and $p^{\prime }(c)\ne 0$ then $p$ has at least two roots

• A non-constant polynomial is irreducible over a field $F$ if its coefficients belong to $F$ and it cannot be factored into the product of two non-constant polynomials with coefficients in $F$.

• A polynomial $D$ can be factored as a product of linear factors (of the form $ax+b$) and irreducible quadratic factors (of the form $a{x}^2+bx+c$, where $b^2-4ac<0$)

• A quadratic polynomial with no real roots is called irreducible over the real numbers. Such a polynomial cannot be factored without using complex numbers.

• Every polynomial with real coefficients can be factored into a product of linear and irreducible quadratic factors with real coefficients

• A polynomial $Q$ can be factored as a product of linear factors (of the form ax + b) and irreducible quadratic factors (of the form $a{x}^2+bx+c$, where $b^2-4ac<0$)