Question

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

3, -13, and 5 + 4i

A. f(x) = x4 - 8x3 - 12x2 + 400x - 1599

B. f(x) = x4 - 200x2 + 800x - 1599

C. f(x) = x4 - 98x2 + 800x - 1599

D. f(x) = x4 - 8x3 + 12x2 - 400x + 1599

Answer 1

Remember, to have zeros 3, -13 and 5+4i, f has factors x-3, x+13 and x-5+14i. Also if 5+4i is a zero, then 5-4i is also a zero.
So at least f should look like: f(x)=(x-3)(x+13)(x-5+14i)(x-5-14i).

Now all you have to do is expand f to see which of A,B,C or D it is!

Answer 2

TYPO: read 4i and -4i instead of 14i ;)

Answer 3

D

Answer 4

I got C

Answer 5

http://www.wolframalpha.com/input/?i=expand+%28x-3%29%28x%2B13%29%28x-5%2B4i%29%28x-5-4i%29

Answer 6

this part \((x-(5+4i))(x-(5-4i))\) can be a bit tricky. there are several ways to do it

Answer 7

one way is to work backwards:
\[x=5+4i\]
\[x-5=4i\]
\[(x-5)^2=-16\]
\[x^2-10x+25=-16\]
\[x^2-10x+41\] is the quadratic

Answer 8

another way to to memorize that if the zeros are \(a+bi\) and \(a-bi\) then the quadratic with those zeros is \(x^2-2ax+a^2+b^2\)

Answer 9

you can then go right to \((x-(5+4i))(x-(5-4i))=x^2-10x+5^2+4^2\)

Answer 10

of course another way it to type it in to wolfram