Help please...

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 2, -4, and 1 + 3i

a. f(x) = x4 - 2x2 + 36x - 80

b. f(x) = x4 - 3x3 + 6x2 - 18x + 80

c. f(x) = x4 - 9x2 + 36x - 80

d. f(x) = x4 - 3x3 - 6x2 + 18x - 80

Question

Help please...

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.  2, -4, and 1 + 3i
 
a.	f(x) = x4 - 2x2 + 36x - 80
 
b.	f(x) = x4 - 3x3 + 6x2 - 18x + 80
 
c.	f(x) = x4 - 9x2 + 36x - 80
 
d.	f(x) = x4 - 3x3 - 6x2 + 18x - 80

anonymous · Answer

a.
Refer to the Mathematica attachment.

whpalmer4 · Answer

The way you would construct the answer is by remembering that such a polynomial will have the roots listed, plus any complex conjugates not listed.  Therefore, our factors will be $$f(x) = (x-2)(x+4)(x-1-3i)(x-1+3i)$$If you multiply that all out, I suggest doing the two complex factors first:
$$f(x) = (x-2)(x+4)(x-1-3i)(x-1+3i)=(x-2)(x+4)(x^2-2x+10)$$$$=(x-2)(x^3+2x^2+2x+40)=x^4+2x^3+2x^2+40x-2x^3-4x^2+-4x-80$$

whpalmer4 · Answer

If you want to play the elimination game, plug x=2 into the various equations and see if you get 0 as the result.  If you have multiple candidates which pass the test, repeat with x=-4.