MHB Polynomial Challenge: Find # of Int Roots of Degree 3 w/ Coeffs

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The discussion revolves around finding the number of degree 3 polynomials with integer coefficients that satisfy the conditions P(0)=3, P(1)=11, and have exactly 2 integer roots. Participants analyze the implications of these conditions on the polynomial's structure and coefficients. The challenge emphasizes the relationship between the roots and the polynomial's values at specific points. The conversation highlights the mathematical reasoning required to determine the possible configurations of such polynomials. Ultimately, the goal is to establish how many distinct polynomials meet these criteria.
anemone
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If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?
 
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beacuse P(0) is odd it does not have any even root and because P(1) is odd it does not have any odd root. So it cannot have any integer roots. So there is no polynomial.

reason : P (a) - P(b) is divisible by a - b
 
the above is based on http://mathhelpboards.com/linear-abstract-algebra-14/polynomal-divisibility-10507.html#post48739
 
Hi kaliprasad,

Thanks for participating and the follow-up explanation post! I also want to thank you for your continuous support to my challenge problems!:)
 
anemone said:
If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?
for convenience we let the leading coefficient=1, then :
$P(x)=x^3+ax^2+bx+3$
$P(1)=1+a+b+3=11$
$\therefore a+b=7$-----(1)
if m,n are 2 intger roots of $ P(x)$ then m.n must be a factor of 3
if $P(-1)=0$ we have $-1+a-b+3=0, \,, =>a-b=-2---(2)$
from (1)(2)
$a=\dfrac {5}{2}$
does not fit (since a must be integer)
if $P(3)=0 $
we have $27+9a+3b+3=0, \,, =>3a+b=-10---(3)$
if $P(-3)=0$
we have $-27+9a-3b+3=0\,, =>3a-b=8----(4)$
from (1)(3) and (1)(4) we find both "a" are not integer
and we conclude such P(x) does not exist
 
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