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My question is:

Find the value of $k$ such that $4x^6 - 24x^5 + 20x^4 + 68x^3 -44x^2 - 40x + k$ is a perfect square.

hey all i have made an edit. Sorry for the inconvenience.

Any help to solve this question would be greatly appreciated.

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    Doesn't matter. It would be worth it to me to have the arithmetic simpler when calculating the early coefficients.2012-06-26

1 Answers 1

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If the goal is to find $k$ so that the polynomial is a perfect square, start by noting that it must be the square of a cubic polynomial:

$4x^6 - 24x^5 + 20x^4 + 68x^3 -44x^2 - 40x + k=(ax^3+bx^2+cx+d)^2\;.$

Clearly this immediately require that $a=2$. Now the square of $(2x^3+bx^2+cx+d)^2$ is

$4x^6+4bx^5+(4c+b^2)x^4+(4d+2bc)x^3+(2bd+c^2)x^2+2cdx+d^2\;,$

so you have the following system of equations:

$\left\{\begin{align*} &4b=-24\\ &4c+b^2=20\\ &4d+2bc=68\\ &2bd+c^2=-44\\ &2cd=-40\\ &d^2=k \end{align*}\right.$

Clearly $b=-6$; the second equation then allows you to find $c$, and you can then use the third, fourth, or fifth to find $d$ and then $k$. (To play safe, you should verify that the third, fourth, and fifth equations all yield the same value of $d$.)

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    @DayLateDon: Well, it *was* the only interpretation that made much sense, given the title! (And the fact that the numbers work out so nicely makes it even more likely, I think.)2012-06-26