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

Algebra ->  Polynomials-and-rational-expressions -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
You have it right. Just look at your zeros, 1-sqrt%2813%29 and 1%2Bsqrt%2813%29 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 x-a is a factor of the polynomial, so for your problem the factors are:
x-%281-sqrt%2813%29%29 and x-%281%2Bsqrt%2813%29%29, so:

y=%28x-1%2Bsqrt%2813%29%29%28x-1-sqrt%2813%29%29

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.

y=x%5E2-x-x%2Asqrt%2813%29-x%2B1%2Bsqrt%2813%29%2Bx%2Asqrt%2813%29-sqrt%2813%29-13

Now collect terms:
y=x%5E2-2x-12. Done.

To check the answer, you can complete the square on the resultant polynomial:
Set the polynomial equal to 0:
x%5E2-2x-12=0

Put the constant on the right:
x%5E2-2x=12

Divide the first degree term coefficient by 2 and square the result, adding the result to both sides:
x%5E2-2x%2B1=13

%28x-1%29%5E2=13

x-1=sqrt%2813%29 or x-1=-sqrt%2813%29

x=1%2Bsqrt%2813%29 or x=1-sqrt%2813%29. Answer checks.