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Find a degree 4 polynomial having zeros -6, -3, 2 and 6 and the coefficient of $x^4$ equal 1

The first step is something like $p(x)=c(x-6)(x+3)(x-2)(x-6)$ as those are all the 4 zeros.

The coefficient of $x^4$ equal to 1 is throwing me off though.

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Try multiplying out your polynomial, treating $c$ as an unknown constant for now. What does the coefficient of $x^4$ become? How can you make it $1$ by choosing $c$ appropriately? – Henning Makholm Nov 5 '11 at 4:57

2 Answers 2

You are almost right and almost done. That first factor should be $x+6$, not $x-6$ (which you have twice). Any value of $c\neq 0$ will give you a polynomial with the roots in the right place.

And when you multiply out what you have, $$c(x+6)(x+3)(x-2)(x-6) = cx^4 + (\text{lower terms}).$$ So if you want the coefficient of $x^4$ to be $1$, then $c$ should be...

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Thanks guys! I feel pretty stupid because i was almost certain that there was another step needed beyond the c(x+6)(x+3)(x-2)(x-6). – Joey O Nov 5 '11 at 5:03

You have it correct, just change the minus to plus on the first (x-6) term and set c=1, multiplying the expression out would give you: $x^4+...$ which is what you need

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