SOLUTION: Directions: Find a polynomial of minimum degree (there are many) that have the given zeros: 1-sqrt(13), 1+sqrt(13). I understand that if the zeros are: -2,0,2 then the polyn

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Question 131842This question is from textbook College Algebra
: Directions: Find a polynomial of minimum degree (there are many) that have the given zeros: 1-sqrt(13), 1+sqrt(13).
I understand that if the zeros are: -2,0,2 then the polynomial would work out like this: x(x+2)(x-2)=x(x^2-4)=x^3-4x
But the whole square root thing is really throwing me off, if you could help, I would really appreciate it. Thank you for your time. :o) This question is from textbook College Algebra

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!
You have it right. Just look at your zeros, and as if they were just numbers (which they are actually), and do what you would do with a number.

a is a root of the polynomial equation if and only if is a factor of the polynomial, so for your problem the factors are:
and , so:

Now you have to extend the FOIL process somewhat to multiply what are actually trinomials, but if you follow a pattern, it will all work out.

Now collect terms:
. Done.

To check the answer, you can complete the square on the resultant polynomial:
Set the polynomial equal to 0:

Put the constant on the right:

Divide the first degree term coefficient by 2 and square the result, adding the result to both sides:

or

or . Answer checks.