polynomials

Division Algorithm
If $F(x)$ and $G(x)$ are polynomials, there exist unique polynomials $Q(x)$ and $R(x)$ such that

$F(x)=Q(x)G(x)+R(x),$

whether (i) $R(x)$ is the zero polynomial or (ii) deg $R$ < deg $G$
Remainder Theorem
When a polynomial $F(x)$ is divided by $x-a$ , the remainder is $F(a)$ .
Factor Theorem
$x-a$ is a root of the equation $F(x)=0$ if and only if $x-a$ is a factor of $F(x)$ .

The Remainder and Factor Theorems are easy consequences of the Division Algorithm. By setting $G(x) = x - a$ and applying the Division Algorithm, we see that $R(x)$ is constant regardless of whether (i) or (ii) holds. Substituting $x = a$ into $F(x) = Q(x)(x - a) + R(x)$ , we see that the constant value of $R$ is $F(a)$ . It follows that if $F(a) = 0$ then $x - a$ is a factor of $F(x)$ . Conversely, if $x - a$ is a factor of $F(x)$ then $F(a) = 0$ .

Problem. Find the unique polynomial $P$ of degree three that satisfies $P(0) = 0$ and $P(1) = P(2) = P(3) = 1$ .
Solution. According to the Factor Theorem, $P(x)-1$ has factors
$x-1$ , $x-2$ , and $x-3$ . It follows that there is a constant $C$ such that

$P(x) = 1 + C(x - 1)(x- 2)(x - 3).$