SOLUTION: Let f(x) be a polynomial with integer coefficients. There exist distinct integers p, q, r, s, t such that f(p) = f(q) = f(r) = f(s) = 18 and f(t) > 18. What is the smallest p

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Let f(x) be a polynomial with integer coefficients. There exist distinct integers p, q, r, s, t such that f(p) = f(q) = f(r) = f(s) = 18 and f(t) > 18. What is the smallest p      Log On


   

Question 1209695: Let f(x) be a polynomial with integer coefficients. There exist distinct integers p, q, r, s, t such that
f(p) = f(q) = f(r) = f(s) = 18
and f(t) > 18. What is the smallest possible value of f(t)?
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 22

Explanation
I'm following a similar pathway discussed on this page
https://math.stackexchange.com/questions/1041011/a-polynomial-fx-and-its-behavior-as-ft5
f(p) = 18
f(p)-18 = 0
g(x) = f(x)-18 = 0
We see that x = p is a root of g(x)
This makes (x-p) a factor of g(x).

You should find that (x-q), (x-r), and (x-s) are also factors of g(x) using similar logic.

What's also a factor is the unknown polynomial h(x)
It might be equal to 1, or it might be something far more complicated.
The only thing we know about h(x) is that it produces an integer output for any integer x; otherwise g(t) wouldn't be an integer and f(t) wouldn't be an integer either.

So far we have
g(x) = (x-p)(x-q)(x-r)(x-s)h(x)
which is the same as saying
f(x)-18 = (x-p)(x-q)(x-r)(x-s)h(x)
f(x) = (x-p)(x-q)(x-r)(x-s)h(x)+18
f(t) = (t-p)(t-q)(t-r)(t-s)h(t)+18

If h(t) = 0, then f(t) = 18
But we must require that f(t) > 18.
This makes h(t) nonzero.

Let's say h(t) = 1 since it's the next integer up.
We want to make h(t) as small as possible.
Same goes for the other factors of (t-p)(t-q)(t-r)(t-s)h(t)
We want h(t) to be positive as the side note at the bottom mentions.

p,q,r,s,t are distinct integers.
This means (t-p), (t-q), (t-r), (t-s) are also distinct integers.
Each must be nonzero or else we run into f(t) = 18 again.
Also it's impossible for 0 to show up since it would imply t is equal to either p,q,r or s.
Select the four smallest nonzero integers {-2, -1, 1, 2} to represent the factors of (t-p)(t-q)(t-r)(t-s) in any order you choose.
The product of those integers is 4.
(t-p)(t-q)(t-r)(t-s) minimizes to 4 when selecting integer inputs and none of the factors can be zero.

So we'd have f(t) bottom out at 4*1+18 = 22 which is the final answer.

Side note: we cannot select h(t) = -1 since it would produce f(t) = 4*(-1)+18 = 14, which violates the condition f(t) > 18.
This is why h(t) cannot be negative.