Question

What is the simplest polynomial function that can be written with zeros 4+1 and ± √2 ?

Answer 1

some weird cut and past here

Answer 2

it is maybe \(4+i\) and \(\pm\sqrt{2}\) ?

Answer 3

it doesn't like you copying the math written as an object

Answer 4

@satellite73 yes fancy italic i. I'm not asking anyone to do my homework. i just don't know how to figure this out at all

Answer 5

so it is definitely \(4+i\) right?

Answer 6

@satellite73 and ±2√

Answer 7

you are going to have a big of work to do, but most of it is not too hard
if the zeros are \(\pm\sqrt2\) then you know it factors as
\[(x-\sqrt{2})(x+\sqrt{2})=x^2-2\]

Answer 8

hmm am confused about the \(\pm2\sqrt{}\) that does make sense to me

Answer 9

could it have been \(\pm\sqrt2\) instead? that i understand

Answer 10

@satellite73 wow i can't type at all today. the question states and ±√2.

Answer 11

oh good
now we can get somewhere

Answer 12

from those two zeros , you know it has the factors
\[(x-\sqrt2)(x+\sqrt2)\] and when you multiply them out, you get
\[x^2-2\] so far so good?

Answer 13

we are not done, i am just asking if that step is okay so far

Answer 14

okay, that makes sense.

Answer 15

now we have to come up with a polynomial that has a zero of \(4+i\) which will also have a zero of \(4-i\) the conjugate
there are many ways to do this
one of the simpler ones is to set
\[x=4+i\] and work backwards

Answer 16

here goes
\[x=4+i\] subtract \(4\) get
\[x-4=i\] then square both sides to get
\[(x-4)^2=i^2\] or
\[x^2-8x+16=-1\] finally add \(1\) to get
\[x^2-8x+17\] as your polynomial with zero \(4+i\)

Answer 17

you last job is to multiply
\[(x^2-2)(x^2-8x+17)\] to get your answer in standard form

Answer 18

@satellite73 x^4−8x^3+15x^2+16x−34 ?

Answer 19

if you have to do a lot of these it is easiest just to memorize that if \(a+bi\) is the zero of a quadratic polynomial then the polynomial is
\[x^2-2ax+(a^2+b^2)\]

Answer 20

looks good to me
wanna check it?

Answer 21

@satellite73 thank you so much!

Answer 22

here you can check that it has the right zeros
http://www.wolframalpha.com/input/?i=x^4%E2%88%928x^3%2B15x^2%2B16x%E2%88%9234+

Answer 23

yw, now i have a question

Answer 24

who is "gisa" that should drop dead?

Answer 25

or is that a play on "drop dead gorgeous" ?

Answer 26

@satellite73 i made this account when I was going through my "Scene emo" stage and it sounded realllllly cool.. at the time.

Answer 27

emo
nice
favorite was "if arsenic fails, try algebra"