I had the task of writing $f = x^4 + 3 x^3 − 35 x^2 − 39 x + 70 \in \mathbb{Q}[x]$. as a linear factorization by guessing the first two null points.
I found $x=-2$ and $x=1$ and did a polynomial division accordingly. Now, for the last two zero points I've calculated $x=-7$ and $x=5$ with the $pq$-formula.
Now my question: Can I now directly write $f=(x+7)(x-5)(x+2)(x-1)$ or do I have to prove that I can do the polynomial division with $f$? Also under the task there was the hint that
You can take advantage of the fact that an integer zero of a polynomial is a divisor of the absolute term.
Does the hint change anything about the need to prove what I mentioned above?