Question

determine an equation in simplified form for the family of quartic functions with zeros -1-√5 and -1+√5 and 2+√2 and 2-√2

Answer 1

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Answer 2

We first find a quadratic equation with \[-1-\sqrt{5}\] and \[-1+\sqrt{5}\] as roots. We find the sum and product of the roots.
Sum of roots: - 2
Product of roots: - 4
Now, we reverse the sign of the sum of the roots. That is 2.
The equation is \[x^2 +2x-4=0\].

For the second quadratic equation.
The equation must have \[2+\sqrt2 \] and \[2-\sqrt2\] as roots.
Sum of roots: 4
Product of roots: 2
Again, invert the sign of the sum of roots. That is - 4.
The equation is \[x^2 -4x+2=0\].
Multiply: \[(x^2 + 2x - 4)(x^2 -4x + 2) = 0\].
Then, the result is your final answer.
Hope this helps a lot :)