Root
Posted: June 11th, 2020, 10:27 am
by 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
Re: Root
Posted: June 11th, 2020, 11:21 am
by UncleEbenezer
cinelli wrote: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.
Re: Root
Posted: June 14th, 2020, 10:33 am
by cinelli
UncleEbenezer wrote:... 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
Re: Root
Posted: June 14th, 2020, 11:10 am
by 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.
Re: Root
Posted: June 14th, 2020, 12:20 pm
by 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