Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

4, -8, and 2 + 5i

Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

4, -8, and 2 + 5i

cruffo · Answer

Since 2 + 5i is a zero, then 2 - 5i is also a zero.  Thus, since there are four zeros, the minimum degree will be 4, that is, the polynomial will be of the form $P(x) = ax^4 + bx^3 + cx^2 +dx + e$

anonymous · Answer

ok...but i cant get any of the answers that are the choices..:(

cruffo · Answer

Yah, there is still a bit of work to do...
You have to get the factors of the polynomial from the zeros, then multiply all the factors together to get the polynomial.

cruffo · Answer

Now, to get the factors from the zeros :
Since x = 4 is a zero, then x - 4 = 0  so (x - 4) is a factor.
Since x = -8 is a zero, then x + 8 = 0, so (x + 8) is a factor.
Since x = 2 +5i is a zero, then x - 2 - 5i = 0, so (x - 2 - 5i) is a factor.
Since x = 2 +5i is a zero, then x - 2 - 5i = 0, so (x - 2 - 5i) is a factor.
Since x = 2 - 5i is a zero, then x - 2 + 5i = 0, so (x - 2 + 5i) is a factor.

So the polynomial can be written as 
$$P(x) = (x - 4)(x - 8)(x-2-5i)(x - 2 + 5i)$$

anonymous · Answer

f(x) = x4 - 6x3 - 20x2 + 122x - 928

 f(x) = x4 - 19x2 + 244x - 928

 f(x) = x4 - 6x3 + 20x2 - 122x + 928

 f(x) = x4 - 61x2 + 244x - 928

anonymous · Answer

those are the choices

cruffo · Answer

Now, to multiply the factors (we can do this in pairs):
$$(x-4)(x+8) = x^2 + 4x - 32$$
and 
$$(x-2-5i)(x-2+5i) = x^2-4x +25$$

So $$P(x) = (x^2+4x-32)(x^2-4x+25)$$

cruffo · Answer

To finish, just multiply these two trinomials, gather liker terms, and you should get one of the answer choices.

cruffo · Answer

I'm not getting any of the answer choices...

anonymous · Answer

x4 - 4x3 + 25x2 + 4x3 - 16x2 + 100x - 32x2 + 128x - 800

x4 - 23x2 + 228x -800 is what i get

anonymous · Answer

what did you get? idk how this is getting so wrong

cruffo · Answer

I made a mistake when I multiplied $(x-2-5i)(x-2+5i)$. I should have gotten $x^2 - 4x + 29$. http://www.wolframalpha.com/input/?i=factor+++x%5E4+-+19x%5E2+%2B+244x+-+928

anonymous · Answer

so its x^4 - 19x^2 + 244x - 928

anonymous · Answer

???

cruffo · Answer

Yes.  That is correct.  Sorry for the delay.