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

Complex zeros come in conjugate pairs. Do you know what the conjugate zero of -2i is?

Answer 2

Is it 2i?? I'm not really sure.... :/

Answer 3

That's correct! If you have a complex number \(a+bi\), the conjugate will be \(a-bi\). Here we've got \(b = \pm2\) and \(a = 0\).

A polynomial with \(n\) zeros \(z_1, z_2, ... z_n\) can be written as a product of binomials like this:
\[(x-z_1)(x-z_2)(x-z_3)...(x-z_n) =0\]

We know that two of the terms are \[(x-2i)(x+2i)\]If we multiply that expression out, we get \[x^2-2ix+2ix-4i^2= x^2 + 4\]. Divide our original polynomial by that and you'll have a new polynomial which if factored will give us the other zeros. Do you know how to divide polynomials?

Answer 4

No... I'm sort of confused! :P

Answer 5

Do you agree that if we can figure out how to divide\( x^4 - 45x^2 - 196\) by \(x^2 + 4\) we'll have a polynomial with just the remaining 2 zeros?

Answer 6

Yes.... So, how would you do that?

Answer 7

Just like you would plain old long division, it turns out.

What is the biggest factor you can multiply with \(x^2\) without going over \(x^4\)?

Answer 8

x^3??????

Answer 9

\[x^2*x^3 = x^5\] doesn't it?

Answer 10

oh, so, x^2?

Answer 11

Right! So what is x^2 * (x^2 + 4)?

Answer 12

x^4 + 4x^2? (:

Answer 13

Exactly. So what is\[( x^4 - 45x^2 - 196 ) - (x^4 + 4x^2)\]

Answer 14

-41x^2 -196 ?? I feel like I did something wrong! :P

Answer 15

I'm sorry, but, I have to go! D: Here are my answer choices:
2i, 14i, -14i

2i, 7, -7

2i, 14, -14

2i, 7i, -7i

Could you please help me find the answer? (:

Answer 16

Close, but not quite.
\[x^4 - 45x^2 - 196 - x^4 - 4x^2\]Collect like terms
\[-49x^2 - 196\]Now, are there any common factors there? Hint: there are :-)

Answer 17

\[-49(x^2 + 4) = -49x^2 - 196\]

So our division yielded\[ (x^2 -49)\] as the quotient. If you factor that, you'll get the other two zeros.

Answer 18

If we had written it like a long division problem, it might look like this:

x^2 - 49
--------------------
x^2 + 4 | x^4 - 45x^2 - 196
-( x^4 + 4x^2 )
---------------
0x^4 -49x^2 - 196
- ( -49x^2 - 196 )
--------------------
0

Answer 19

Hint for factoring the rest:
\[(a-b)(a+b) = a^2 -ab + ab - b^2 = a^2 - b^2\]