7. Some useful properties of polynomials

Except as noted, it will be assumed that polynomials have real numbers as coefficients, rather than any complex numbers. Some of these properties will be useful for the Exercises; others are listed for the sake of completeness.

• Factor corresponding to a root. If $f(t)$ is a polynomial and $a$ is a root of $f(t)$, then $(t-a)$ is a factor of $f$.

  (The reason: You can always do a long division of $f(t)$ by $(t-a)$ to get a quotient $q(t)$ and a remainder $r(t)$, which being of lower degree than $(t-a)$ is simply a constant $r$. Thus $f(t) = q(t)(t-a) + r$. Put $t = a$; since $f(a) = 0$ you get $0 = 0 + r$, so $r = 0$. In other words, $f(t) = q(t)(t-a)$, so $(t-a)$ is a factor of $f$.)

  
  
  

• Number of roots. A nonzero polynomial of degree $n$ has at most $n$ distinct roots.

  (The reason: Not more than $n$ different terms $(t - a_i)$ could be factors of $f(t)$, since $f(t)$ is of degree $n$. Actually, this reason requires further explanation: When you have factored out some of the terms $(t - a_i)$ for roots $a_i$ of $f(t)$ and you have an equation $f(t) = (t - a_1)(t - a_2) \dots (t - a_k) q(t)$, you need to observe that the remaining roots of $f(t)$ are also roots of $q(t)$, so you can keep factoring.)

  
  
  

• Expression from function. If $f$ and $g$ are polynomials that are equal as real functions, then they have the same coefficients.

  (In other words, in talking about ``equal polynomials'' we don't have to say whether we mean equal as functions or equal as polynomial expressions. By way of contrast, in modern algebra ones studies polynomials with coefficients in finite fields; in this setting two polynomials with different coefficients can give the same function.)

• Agreement in many places gives agreement everywhere. If two polynomials $f(t)$ and $g(t)$ of degree at most $n$ agree at $n+1$ or more values of $t$ then $f = g$, i.e., $f(t) = g(t)$ for all $t$.

  (The reason: The values of $t$ at which $f$ and $g$ agree are roots of the difference $f(t) - g(t)$. If this difference is nonzero it can have at most $n$ roots.)

  
  
  

• Continuation property. If $f$ and $g$ are polynomials that agree on an interval, then they are the same polyonomial.

  (For example, if $f(t) = g(t)$ for $0 \leq t \leq 0.000001$, then $f(t) = g(t)$ for all values of $t$.)

  
  
  

• Unique interpolation property (Lagrange property). There is exactly one polynomial $f(t)$ of degree at most $n$ that has given values at $n+1$ distinct values of $t$ (i.e., $f(t_0) = y_0,\dots, f(t_n) = y_n$).

  (This is just a nonparametric version of Theorem . The uniqueness comes from the agreement property above.)

  
  
  

• Unboundedness: A nonconstant polynomial is unbounded. In other words, if $f(t)$ is nonconstant there is no constant $B$ such that $\vert f(t) \vert \leq B$ for all $t$.

  (The reason: If $f(t) = c_n t^n + \dots + c_1 t + c_0$, with $c_n \neq 0$, for large values of $t$ the leading term $c_n t^n$ overwhelms the other terms and makes the value of $f(t)$ large. More formally, if $\vert c_n t^n + \dots + c_1 t + c_0 \vert \leq B$ for all $t$ with $c_n \neq 0$, consider values of $t$ with $t > 0$ and divide through by $t$ to get $\vert c_n + {{\frac{\displaystyle c_n}{\displaystyle t}}} + \dots + {\frac{\di... ...c_0}{\displaystyle t^n}}\vert \leq {\frac{\displaystyle B}{\displaystyle t^n}}$. Then let $t \rightarrow \infty$. You get $\vert c_n \vert \leq 0$, a contradiction.)

  
  
  

• Existence of real roots for odd degree. Any polynomial of odd degree with real numbers as coefficients must have at least one real root.

  (The reason: Suppose the polynomial is $f(x)$ and its leading coefficient is positive. For a large negative value of $t$, $f(t) < 0$; for a large positive value of $t$, $f(t) > 0$. By continuity, in between somewhere there is a $t$ with $f(t) = 0$. If the leading coefficient is negative, similar reasoning applies.)

  
  
  

• Existence of complex roots. Any nonconstant polynomial, with real coefficients or with even complex coefficients, has at least one complex root.

  This is one statement of the Fundamental Theorem of Algebra. The following fact is another, equivalent statement:

  
  
  

• Complex factorization. If complex numbers are used, any polynomial can be factored completely into linear factors:
  $f(t) = c (t - a_1) (t - a_2)\dots (t - a_n)$.

  
  
  

• Complex roots of real polynomials. If a real polynomial is factored into complex linear factors, the non-real roots occur in conjugate pairs.

  Thus if there is one factor $(t - (3+2i))$, then there must be one factor $(t- (3-2i))$.