What's the intuition behind the fact that you can factorise an expression as $(x-a)(x-b)$ if $a$ and $b$ are the roots of the expression

Question

Say $f(x)$ is a polynomial function with roots $a, \, b, \, c$ then this can be expressed as $f(x) = (x-a)(x-b)(x-c)$

What's the intuition behind this? Why is this true?

Answer 1

The Euclidean algorithm applied to polynomials tells us that for any polynomial $f(x)$ then

$f(x) = (x-a)q(x) + r$

for some polynomial $q(x)$ and some constant $r$. Note that $r$ is a constant because it must have a degree strictly less than the degree of $x-a$, which is $1$.

But if $a$ is a root of $f(x)$ then $f(a)=0$, so $r=0$, and so

$f(x) = (x-a)q(x)$

i.e. $x-a$ is a factor of $f(x)$.