Using the division algorithm repeatedly, show
$$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = a_n (x-k) (x-j) \cdots (x-b)$$ for $n$ greater than or equal to $1$.
My attempt: (Proof by induction) Consider the case $n=1$. Then, we can write $$ax + b=a\left(x-\left(-\frac{b}{a}\right)\right).$$ Now assume it is true that $$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = a_n (x-k) (x-j)\cdots(x-b)$$ for some constants $k,j,\ldots,b$.
We will show that $$a_{n+1} x^{n+1} + a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = a_{n+1}(x-k')(x-j')\cdots(x-b')$$ for some constants $k',j',\ldots,b'$.
I am stuck here, do I use the division algorithm here?