Question

Using the given zero find all other zeros of f(x) .
-2i is a zero of f(x)=x^4-45x^2-196

Answer 1

If -2i is one zero, what do you know must be another zero? (From the conjugate zeroes theorem.)

Answer 2

that the conjugate of it will be another one or something?

Answer 3

The conjugate of
a + bi
is
a - bi

So what's the conjugate of -2i?

Answer 4

positive 2i?

Answer 5

@agent0smith xD

Answer 6

Yes, so that means you have
(x+2i)(x-2i) as factors.

Expand that out,

then use polynomial long division with the original function being divided by that result

Answer 7

Ok let me try this outttt :D

Answer 8

Well good luck because I'm probably not going to show you how to do the long division :P Not that easy to do on here, so I hope you remember how to do it.

Answer 9

okay this is what I did.

x^2-2ix+2ix-4i^2, is that good? I want to make sure before I divide

Answer 10

and can I do synthetic division instead of the other way?

Answer 11

@agent0smith

Answer 12

You can't do synthetic division unless you're dividing by linear factors, that's a quadratic.

And no, simplify x^2-2ix+2ix-4i^2 before you divide.

Answer 13

oh okay, completely simplify it like (x+2) (x+2), or just like x^2-4i^2?

Answer 14

x^2-4i^2 simplifies to...?

Answer 15

x^2+4?

Answer 16

Yes. Now divide.

Don't forget x^4-45x^2-196 is missing an x^3 and and x term, so you need to have zeroes in there like x^4+0x^3-45x^2+0x-196 before dividing.

Answer 17

You can also just factor this. x^4-45x^2-196 is like a quadratic. Replace x^2 with u to make it easier.

u^2 - 45u - 196

Now find two numbers that multiply to -196 and add up to -45. You already know one from the above x^2+4, is 4, so the other is -49

(u+4)(u-49)

now replace the u with x^2 again

(x^2+4)(x^2-49)

Now factor that second set of brackets, as it's a diff of two squares.

Answer 18

so it will be postive 7 and negative 7 then?

Answer 19

Yes

Answer 20

AW YES LAWD. Thank you so much bro! :D

Answer 21

lol no prob