A Strange Polynomial in 2015

Let $g(x) = f(x) - n$ . Some of the roots of $g(x)$ occur when $f(x) = n$ i.e. at $x = 2, 0, 1, 5.$ Thus, $g(x)$ can be factored as $x(x-1)(x-2)(x-5)h(x)$ for some polynomial $h(x)$ . If we plug in $-2, -1, -5$ into $f(x) - n,$ we get

$\begin{aligned} f(-2) - n &= g(-2) = (-2)(-3)(-4)(-7)h(-2) \\ &= (2^3 \times 3 \times 7)h(-2) = -2n \\ \\ f(-1) - n &= g(-1) = (-1)(-2)(-3)(-6)h(-1) \\ &= (2^2 \times 3^2)h(-1) = -2n \\ \\ f(-5) - n &= g(-5) = (-5)(-6)(-7)(-10)h(-5) \\ &= (2^2 \times 3 \times 5^2 \times 7)h(-5) = -2n \\ \end{aligned}$

Thus, $-n = (2^2 \times 3 \times 7)h(-2) = (2 \times 3^2)h(-1) = (2 \times 3 \times 5^2 \times 7)h(-5).$ The smallest value of $n$ that satisfies this is the least common multiple of $2^2 \times 3 \times 7$ , $2 \times 3^2$ , and $2 \times 3 \times 5^2 \times 7$ . This is $2^2 \times 3^2 \times 5^2 \times 7 = \boxed{6300}.$

For the sake of completeness, we show that a polynomial $f(x)$ with our value of $n$ does exist. Plugging into our last equation, we get $h(-2) = -75$ , $h(-1) = -350$ , and $h(-5) = -6$ . Using Lagrange, we get $h(x) = -63x^2 -464x - 751$ . Thus,

$\begin{aligned} f(x) - 6300 &= x(x-1)(x-2)(x-5)h(x) \\ &= x(x-1)(x-2)(x-5)(-63x^2-464x-751) \\ &= -63x^6 + 40x^5 +1890x^4 -1250x^3 -8127x^2 +7510x, \end{aligned}$

or $f(x) = -63x^6 + 40x^5 +1890x^4 -1250x^3 -8127x^2 +7510x + 6300$ .