Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that$$f(p) = f(q) = f(r) = f(s) = 5.$$If $t$ is an integer and $f(t)>5$, what is the smallest possible value of $f(t)$?
Let f(x) be a polynomial with integer coefficients.
Suppose there are four distinct integers p,q,r,s such that f(p) = f(q) = f(r) = f(s) = 5. (Can there be more??)
If t is an integer and f(t)>5,
what is the smallest possible value of f(t) ?
I'd like to see this one done too, it makes little sense to me .
Conversation between friend and me:
So you can write f(x) = (x-p)(x-q)(x-r)(x-s)+5, which will satisfy the thing and then minimize it since everything is integer
but how to minimize it]
isnt the minimumn 5?
Plug in x=t
Basically minimize positive value of four different integers (t-whatever)
Value of product of four
f(t) = (t-p)(t-q)(t-r)(t-s)+5>5, so (t-p)(t-q)(t-r)(t-s)>0
(t-p)(t-q)(t-r)(t-s) would be 1*2*3*4?
Yeah so the product of the four t's need to be positive and distinct since p,q,r,s are different
so the answer wold be 24?
The product has to be positive but not each term
And add 5
-1, 1, -2, 2 product=4, plus 5 is 9?
so it would be 9?
And the answer is 9