Question

write a polynomial function that has the given zeros
2, 8i
i know how to get the (x-2) but I dont know ow to deal with the 8i. HELP!

Answer 1

x = 8i
square both sides:
x^2 = 8^2 * i^2 = 64 * (-1) = -64
x^2 + 64 = 0. Therefore, (x^2+64) is one factor.

x = 2
x - 2 = 0. Therefore, (x - 2) is another factor.

Multiply the two factors.

Answer 2

hmm, must say it never occured to me to go that route on the imaginary roots

Answer 3

Saves so much hassle. If one of the complex roots were: 2 + 3i then
x = 2 + 3i
x - 2 = 3i
(x-2)^2 = (3i)^2 = -9
x^2 - 4x + 4 + 9 = 0
x^2 - 4x + 13 = 0 has the roots 2 + 3i and 2 - 3i.

Answer 4

i like it ... might not remember it after awhile ... but i do like it :)

Answer 5

omg I really don't get it. why is it x^2?? I kind of see where you are coming from but not completely.

Answer 6

We square both sides to get rid of the i.

Answer 7

You can do it the long way too.
Remember that complex roots always occur in conjugate pairs.
So if 8i is one complex root, then its conjugate, -8i is also a root.
So, (x - 8i) and (x + 8i) are factors.
Multiply them while making use of the identity (a+b)(a-b) = a^2 - b^2

Answer 8

(x + 8i) * (x - 8i) = 0
x^2 - (8i)^2 = 0
x^2 - (-64) = 0
x^2 + 64 = 0 (x^2 + 64) is one factor.

Answer 9

(x - 2) is another factor.

So the polynomial is: (x - 2)(x^2 + 64) = x^3 + 64x - 2x^2 - 128
x^3 - 2x^2 + 64x - 128

Answer 10

okay so if i had to wirte a polynomial function with the zeros -1, 3i,4
i would do (x+1)(x^2+9)(x-4) ??

Answer 11

Exactly! Multiply it out and arrange the polynomial in decreasing powers of x.

Answer 12

uhm okay so from (x+1)(x^2+9)(x-4) i would go to (x^3+9x+x^2+9)(x-4) ?

Answer 13

yes.

Answer 14

omg! i cant believe im actually getting this ! haha okay so the answer would be x^4+9x^2+3^3+9x-4x^3-36x-4^2-36

Answer 15

and from there, put it in order?

Answer 16

Combine like terms and put it in order.

Answer 17

x^4-3x^3+5x^2-27x-36 ??

Answer 18

Correct.

Look under Real Solutions and Complex solutions for the above equation and it confirms your polynomial is correct: http://www.wolframalpha.com/input/?i=x^4+-+3x^3+%2B+5x^2+-+27x+-+36+%3D+0

Answer 19

thank you so much! i really appreciate it.

Answer 20

You are welcome. You did really good!