Question

F(x) is a fourth-order polynomial with integer coefficients and no common factor among the coefficients. The roots of F(x) are -2, -1, 1 and 2. If p is a prime number greater than 97, then the largest integer that divides F(p) for all such values of p is:

• A. 72
• B. 120
• C. 240
• D. 360

Hint:

Write F(p) as a product of four factors around p, then check what power of 2, 3 and 5 is guaranteed every time.

Answer 1

The Correct Option is D

Since F(x) has roots $-2,-1,1,2$ with integer coefficients and no common factor, $F(x) = (x+2)(x+1)(x-1)(x-2) = (x^2-1)(x^2-4)$, so $F(p) = (p-2)(p-1)(p+1)(p+2)$.

Instead of working out the divisibility rules algebraically, just plug in two actual primes bigger than 97 and take the highest common factor of the results, since the answer has to divide F(p) for every such prime.

Take $p = 101$. Then $F(101) = 99 \times 100 \times 102 \times 103$. Breaking these into primes: $99 = 3^2 \times 11$, $100 = 2^2 \times 5^2$, $102 = 2 \times 3 \times 17$, and $103$ is itself prime. So $F(101) = 2^3 \times 3^3 \times 5^2 \times 11 \times 17 \times 103$.

Take $p = 103$. Then $F(103) = 101 \times 102 \times 104 \times 105$. Breaking these down: $101$ is prime, $102 = 2 \times 3 \times 17$, $104 = 2^3 \times 13$, $105 = 3 \times 5 \times 7$. So $F(103) = 2^4 \times 3^2 \times 5 \times 7 \times 13 \times 17 \times 101$.

The highest common factor of $F(101)$ and $F(103)$ only keeps the smaller power of each shared prime: $2^{\min(3,4)} = 2^3$, $3^{\min(3,2)} = 3^2$, $5^{\min(2,1)} = 5^1$. No other prime, like 11, 13 or 17, is common to both, since those come from the specific p chosen and are not guaranteed for every prime.

So the largest number that must divide F(p) for every large prime p is $2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$.

Checking with a third prime, $p=107$, gives $F(107)=105 \times 106 \times 108 \times 109 = (3 \times 5 \times 7)(2 \times 53)(2^2 \times 3^3)(109)$, which equals $2^3 \times 3^4 \times 5 \times 7 \times 53 \times 109$, still a multiple of 360 and nothing more restrictive, confirming 360 is exactly right.