TITLE: Find coefficients of polynomial, knowing its roots are consecutive integers
QUESTION [5 upvotes]: In the function $f(x)= x^3-15x^2+ax+b$ the graph has $3$ consecutive points where it crosses the x-axis. 
These $3$ points are consecutive integers. Find $a$ and $b$ for this is you know that $a$ and $b$ are real numbers. 
How do I start to find the answer?

REPLY [9 votes]: OK, so let's assume you never heard of Vieta's formulas.
Points where the graph crosses the $x$ axis are roots of the polynomial, and if you know the roots of the polynomial, call them $A, B, C$, then the polynomial can be written as $f(x) = (x-A)(x-B)(x-C)$. Are you supposed to know this much? If not, I would suggest the problem is unfair. So let's say you are allowed to assume this much.
Then $f(x) = x^3 - (A+B+C) x^2 + \cdots$ (or $-\cdots$). So this means $A+B+C = 15$. This is the Vieta formula for the sum of roots, even if you never heard of it (and to solve the problem, you don't need to know it even has a name).
Now $A,B,C$ are consecutive integers, so the middle one must be 5, and the others are 4 and 6. Plug back into $f(x)=(x-4)(x-5)(x-6)$ and find $a, b$ by brute force (multiply out the RHS).