4.6 The factor and remainder theorems

The remainder theorem states that for any number $a$, the remainder when we divide a polynomial (which can be written as $f(x)$) by $(x-a)$ is equal to $f(a)$. ⁶⁶ 6 We can prove this without too much trouble. Firstly, by the definition of division, we can write the result of any division as a ”quotient” $Q(x)$ and a remainder $R(x)$ as a fraction of $x-a$ - as an equation, that is, that $\frac{f(x)}{(x-a)}=Q(x)+\frac{R(x)}{x-a}$. If we multiply both sides by $x-a$, then we obtain that $f(x)=Q(x)(x-a)+R(x)$. We can then set $x=a$, which gives that $f(a)=R(a)$. Note that $Q(x)(a-a)=0$.

A special case of the remainder theorem is known as the "factor theorem" and it relates the roots of a polynomial to its factors. If $(x-a)$ divides $f(x)$ with no remainder, then by the remainder theorem, this means that $f(a)=0$ and so $a$ is a root as well as a factor. Roots are factors, and factors are roots.