Question

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

Answer 1

Can you write a polynomial function of minimum degree and Real coefficients that has 2 as its ONLY zero?

Answer 2

im not understanding what youre asking

Answer 3

if \[1+3i\] is a zero, there must be another one
do you know it?

Answer 4

the question is not that hard, if you know what to do

Answer 5

If there is a zero of "2", there is a factor of (x-2).
If there is a zero of "-2", there is a factor of (x-(-4)) = (x+4).

That's all that part is. Then there is what satellite asked.

Answer 6

so is the other one 1-3i

Answer 7

Yes, and so you have at minimum four roots: 2, -4, 1+3i and 1-3i.

How would you turn these roots into factors of the desired polynomial?

Answer 8

it would be x-2 and x+4 I think?

Answer 9

those are two factors, yes

Answer 10

ow your harder (but that that hard) job is to find the one with zeros \(1\pm 3i\)
do you know how to do that?
there is a hard way, an easy way, and a real real easy way
you pick

Answer 11

i dont know how to do that

Answer 12

ok one way is to work backwards

Answer 13

put \[x=1+3i\] like you just solved a quadartic equation by completing the square
the previous step would be \[x-1=3i\]

Answer 14

squaring both sides gives \[(x-1)^2=(3i)^2\\
(x-1)^2=-9\]

Answer 15

expand to get \[x^2-2x+1=-9\] add 9 to get your quadratic \[x^2-2x+10=0\]

Answer 16

that's the answer?

Answer 17

the other way is to memorize that if one root is \(a+bi\) then the quadratic is \[x^2-2ax+(a^2+b^2)=0\]

Answer 18

that is not your final answer

Answer 19

your final answer is \[(x+4)(x-2)(x^2-2x+10)\]

Answer 20

or whatever you get when you multiply that mess out

Answer 21

ok so i just need to distribute all of that and i should get the answer?

Answer 22

@satellite73 can you help me multiply it out? i got -x^3 + 5x^2 + 2x + 32 but that isnt an answer choice

Answer 23

answer choices are
A. x^4 - 2x^2 +36x - 80
B. x^4 - 3x^3 + 6x^2 - 18x + 80
C. x^4 - 9x^2 +36x - 80
D.x^4 - 3x^3 - 6x^2 + 18x - 80

Answer 24

where did the negative leading coefficient come from?

Answer 25

\[(x+4)(x-2)(x^2-2x+10)\] is what you need right? multiplying out is donkey work, let the computer do it

Answer 26

yes thats it

Answer 27

or else you have to first multiply \[(x+4)(x-2)\] and see what you get
what do you get?

Answer 28

x^2 + 2x - 8

Answer 29

looks good

Answer 30

now you have to multiply \[(x^2+2x-8)(x^2-2x+10)\]

Answer 31

which is a pain, you have \(3\times 3=9\) multiplications to do, then you have to combine like terms
there is a much easier way

Answer 32

i got x^4 + 16 x

Answer 33

not even close

Answer 34

whats the easier way

Answer 35

scroll down to "alternate form"
http://www.wolframalpha.com/input/?i=(x%2B4)(x-2)(x%5E2-2x%2B10)

Answer 36

thank you!!!