Please help medal and fan will be given.
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

Question

Please help medal and fan will be given.
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

anonymous · Answer

f(x) = x4 - 2x2 + 36x - 80

f(x) = x4 - 3x3 + 6x2 - 18x + 80

f(x) = x4 - 9x2 + 36x - 80

f(x) = x4 - 3x3 - 6x2 + 18x - 80 
these are the answer choices

anonymous · Answer

the first part is easiest
you know if the zeros are $2$ and $-4$ then two factors are $$(x-2)(x+4)$$
then you have to find the quadratic whose zeros are $1\pm3i$

anonymous · Answer

there are a couple ways to do this

anonymous · Answer

yeah I did the first part and got x^2+2x-8 ...... not sure what to do with the second half

anonymous · Answer

one way is to work backwards
$$x=1+3i\
x-1=3i\
(x-1)^2=(3i)^2=-9\
x^2-2x+1=-9\
x^2-2x+10=0$$

anonymous · Answer

so your quadratic is $x^2-2x+10$

anonymous · Answer

okay so now I multiply the one I got by the one you got?

anonymous · Answer

the snap way is to memorize that if $a+bi$ is a zero, the quadratic is $$x^2-2ax+(a^2+b^2)$$ in your case $a=1,b=3$ you get 
$$x^2-2x+1^2+3^2=x^2-2x+10$$

anonymous · Answer

your last job is to multiply 
$$(x-2)(x+4)(x^2-2x+10)$$ without making an algebra mistake
i would cheat

anonymous · Answer

how would you cheat?

anonymous · Answer

http://www.wolframalpha.com/input/?i=%28x-2%29%28x%2B4%29%28x^2-2x%2B10%29

anonymous · Answer

Thank you! Can you help me with one more problem?

anonymous · Answer

Using the given zero, find all other zeros of f(x).
-2i is a zero of f(x) = x4 - 45x2 - 196 

 2i, 14i, -14i

2i, 7, -7

2i, 14, -14

2i, 7i, -7i

anonymous · Answer

Hey I just need help with this problem down here. I have no idea.

anonymous · Answer

you know that $-2i$ is a zero and therefore so is $2i$
one factor is $(x-2i)(x+2i)=x^2+4$

anonymous · Answer

so 
$$x^4 - 45x^2 - 196 =(x^2+4)(\text{some quadratic)}$$ and your job is to find the quadratic

anonymous · Answer

is it (x^2+49)?so the answer would be D?

anonymous · Answer

not too hard for this one
the first term has to be $x^2$ and the last term has to be $49$

anonymous · Answer

yeah that is what i get

anonymous · Answer

okay awesome you are the best!

anonymous · Answer

course you can always solve 
$$ x^4 - 45x^2 - 196 $$by treating it as 
$$z^2-45z-196=0$$ where $z=x^2$ and once you have $z$ you can solve for $x$

anonymous · Answer

thanks!