Question

Using the given zero, find all other zeros of f(x).

-2i is a zero of f(x) = x^4 - 5x^2 - 36
These are the answer choices:
(a)2i, 6i, -6i
(b)2i, 6, -6
(c)2i, 3, -3
(d)2i, 3i, -3i

Answer 1

A zero of a function is the value of x when the function =0.
When the function = 0. It means when divided by the factors the remainder =.0

Given -2i is a zero then factor (x + 2i)=o
Complex numbers come in pairs so (x.- 2i) is also a factor

Answer 2

Then what's the answer?

Answer 3

Divide f(x) by the factors

Answer 4

\[\frac{x^4-5x^2-36}{(x-2i)(x+2i)}\]

Answer 5

Example if 2 is a factor of 24. Then to find the other factor divide 24 by 2 = 12. So
2 * 12 = 24
To get all the factors need to break down 12 as well
For example 24 = 2*3*4

Answer 6

what does the i stand for?

Answer 7

The numerator can be factorized Quadratic. Then cancel the common term

Answer 8

i is the square root of negative one

Answer 9

I = square root of -1

Answer 10

I squared = -1

Answer 11

\[\frac{x^4-5x^2-36}{(x-2i)(x+2i)}=\frac{x^4-5x^2-36}{(x^2+4)}=\frac{x^4+(4x^2-9x^2)-36}{(x^2+4)}\]
\[\frac{(x^4+4x^2)-(9x^2+36)}{(x^2+4)}=\frac{(x^4+4x^2)}{x^2+4}-\frac{(9x^2+36)}{(x^2+4)}\]
\[x^2-9\]
\[(x+3)(x-3)\]

Answer 12

A negative number can be -1 times the number
To get the square root it would be rt of -1 times the rt of the number
Rt of -1 is denoted I.

Answer 13

I can show you how to multiply (x-2i)(x+2i)

Answer 14

wait, is it C?

Answer 15

Maybe

Answer 16

i think I got it

Answer 17

how would i be able to find all the answers then?

Answer 18

So a 4th order polynomial (x is raised to the 4th power) has 4 roots.

Answer 19

They give you one root: (x+2i)

Answer 20

An important piece of information is that if all of the coefficients are real, then the complex root (x-2i) MUST be accompanied by its complex conjugate, which is just (x-2i).

Answer 21

Now you know 2 roots. You can combine the roots together to make something that doesn't have i in it.

Answer 22

Wait I get it!

Answer 23

It's C!

Answer 24

(x-2i)(x+2i)
=x(x+2i)-2i(x+2i)
=x^2+(2*x*i)-(2*i*x)+2*2*i^2
=x^2+4i^2
=x^2-4
Remember i is the sqrt of -1