Root

[cinelli]

Show that the quartic equation

( )x^4 + ( )x^3 + ( )x^2 + ( )x + ( ) = 0

where the gaps are filled in by any arrangement of the numbers 1, -2, 3, 4, -6
always has a rational root.

Cinelli

[UncleEbenezer]

Show that the quartic equation

( )x^4 + ( )x^3 + ( )x^2 + ( )x + ( ) = 0

where the gaps are filled in by any arrangement of the numbers 1, -2, 3, 4, -6
always has a rational root.

Cinelli

No maths needed. Just the obvious solution, which is not merely rational but integral, based on the observation that 1 - 2 + 3 + 4 - 6 = 0.

[cinelli]

... based on the observation that 1 - 2 + 3 + 4 - 6 = 0

Therefore ...

Is this explanation incomplete?

I think this is like Poirot saying he knows who did it, but he's not going to tell anyone.

Cinelli

[UncleEbenezer]

You should credit your readers with an ability to see the obvious when it's pointed out.

They can, after all, see the less-obvious with a suitable clue.

[Gengulphus]

I'm with UncleEbenezer about this being obvious - I had the answer within about two seconds of reading the question! But I didn't bother posting it because (a) I happened to read the question over a day after it had been posted; (b) it might more challenging to those who are less mathematically inclined than UncleEbenezer and me...

For anyone who is still trying to solve it, here's a strong hint:

Pick a particular assignment of the numbers 1, -2, 3, 4, -6 to the gaps, e.g. the equation 1x^4-2x^3+3x^2+4x-6 = 0. Spot the very simple value of x that makes that reduce to the true equation 1-2+3+4-6 = 0. Consider whether that value of x will also work for other assignments of the numbers to the gaps.

Gengulphus