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Determine all positive integers $k \le 2000$ for which $x^4 + k$ can be factored into two distinct trinomial factors with integer coefficients

User Micromoses
by
8.6k points

1 Answer

5 votes

The answers are 4, 64, 324, and 1024. A proof is below.

Let the trinomial factors be
ax^2 + bx + c and
dx^2 + ex + f. Then, their product is
(ad)x^4 + (ae+bd)x^3 + (fa+be+cd)x^2 + (bf+ce)x + cf. We'll solve this problem by analyzing each of these coefficients.

Since the
x^4 coefficient is
ad and all of the variables are integers, we have either
a = d = 1 or
a = d = -1. Without loss of generality, let
a=d=1. (In the -1 case, we could simply multiply all the variables by -1 to find an analogous 1 case.)

So, we're back to
x^2 + bx + c and
x^2 + ex + f. We have that the
x^3 coefficient is
b+e, and since that term is zero in this polynomial, we have
b+e=0. This means that
e=-b.

Our polynomials are now
x^2 + bx + c and
x^2 - bx + f. Let's consider the
x coefficient. It shows that we must have
bf - bc = 0, so either
b = 0 or
f - c = 0.

Let's look at the case with
b=0. In this case, our polynomials become
x^2 + c and
x^2 + f. Then, we must have, for the
x^2 coefficient to be zero, that
c+f=0, so
c = -f. But then, considering the constant term, we'd need
-c^2 = k. However, we are given that k is positive, and we know that
-c^2 is negative, so this case doesn't work. Hence, we must have that
f - c = 0, so
f = c.

Our polynomials are thus
x^2 + bx + c and
x^2 - bx + c. Hence,
k = c^2, and since
c is an integer,
k must be a perfect square. Let's look at the
x^2 coefficient again. We must have that
2c - b^2 = 0, so
b^2 = 2c. Hence,
c must itself be half of a perfect square, since
b is an integer.

Let's consider values of
b. Since
b^2 = 2c,
b^2 is even, so
b is even. If
b = 2,
c = 2, and
k = c^2 = 4. If
b = 4,
c = 8, and
k = 64. If
b = 6,
c = 18, so
k = 324. If
b = 8,
c = 32, and
k = 1024. If
b = 10,
c = 50, so
k = 2500: but that doesn't work, because
2500 > 2000. So, our working values for
k are 4, 64, 324, and 1024.

This was a very drawn-out problem, so if you have any questions about how I worked through any of the steps, please feel free to ask!

User Jojie
by
9.0k points
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