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# Integer Roots

-1
218
9
+148

In a certain polynomial, all the coefficients are integers, and the constant coefficient is 42. All the roots are integers, and distinct. Find the largest possible number of integer roots.

Nov 23, 2019

#1
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I think 12 ...

Nov 24, 2019
#3
+108650
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maybe 13.

I äm going with 13, that is till I change my mind again.

Melody  Nov 24, 2019
#2
+1

I think the answer is 3.

Nov 24, 2019
#4
+148
-1

Actually i solved it and got 5 : they would be 2,3,7,1, and lastly -1.

Nov 24, 2019
#5
+108650
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Yes, sounds good.

You found this 5 :        2,3,7,1, and lastly -1.

they could also have been

-2,-3,7,1, and lastly -1.

In fact I think there are  many other groups of 5 roots that would work.

Can you work out how many groups of 5 exist?

Melody  Nov 24, 2019
edited by Melody  Nov 24, 2019
edited by Melody  Nov 24, 2019
#6
+148
-1

I think it's 4 groups.

AoPS.Morrisville  Nov 24, 2019
#7
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+2,+3,+7,  +1, and lastly -1.

-2,-3,+7,   +1, and lastly -1.

-2,+3,-7,   +1, and lastly -1

+2,-3,-7,   +1, and lastly -1.

Looks like you could be right.

Nov 25, 2019
#8
+4450
+1

This is a typical problem for the Rational Root Theorem: Read more here: https://brilliant.org/wiki/rational-root-theorem/

Since the prime factorization of 42=2*3*7, but we can also add the value one(1).

There could be negative values, too. I agree with Melody.

Nov 27, 2019
#9
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Thanks Tetre,

I think that is all of them....

Melody  Nov 27, 2019