Question

Please help. Find all the complex zeros f(x)=x^3-10x^2+44x-69. I don't know what to do after this part below.

Answer 1

well you're doing it correct... once you got zero (0) remainder... next will be a second degree equation which is much easier to solve since we have the quadratic formula for solving the zeros of the function...

Answer 2

So I just keep going until I get zero? What do I do with the remainder I have now?

Answer 3

if you have a remainder, it means it is not a zero of the function, try another factor of 69...

Answer 4

There is no remainder. the last term on the second line should be 69 ( 23 x 3 )

Answer 5

then the resulting numbers of each column would the constants of the 2nd degree or quadratic equation... if 69 is a factor... the resulting factor of the above equation will be...

(x - 69)(ax^2 + bx + c)... where a, b, and c are the result of synthetic divisions you made above...

Answer 6

or is it 3 not 69?

Answer 7

@alekos Oh, I caught that mistake a while ago. But I'm unsure of what I should do after getting x^2-7x+23. How would I find the real zero and complex zeros? Do I use the quadratic formulas?

Answer 8

Yes, just simply use the quadratic formula. Remeber \(\sqrt{-1} = i\)

Answer 9

Okay thanks @Hero

Answer 10

I'm sorry, What's the \[\sqrt{-43}\] ? lol

Answer 11

Recall this square root property:
\(\sqrt{ab} = \sqrt{a}\sqrt{b}\)

In this case, \(a = -1\) and \(b = 43\) and
\(\sqrt{-43} = \sqrt{-1}\sqrt{43}\)

Answer 12

Oh I did that right then. I thought I was wrong, lol. After I've done that, I'm left with \[\frac{ 7\pm \sqrt{-1}\sqrt{43} }{ 2}\] How would that help me find zeros?

Answer 13

As I said before \(\sqrt{-1} = i\)

Answer 14

You use the quadratic formula to find the zeroes of a quadratic function. Once you have simplied it completely, the result represents the zeroes of the function.

Answer 15

I never really understood imaginary #s so I don't understand how I'm supposed to simplify. \[\frac{ 7\pm \sqrt{43i} }{ 2 }\]

Answer 16

Actually, you should have written it as \(\dfrac{ 7\pm \sqrt{43}i }{ 2 }\) or \(\dfrac{ 7\pm i\sqrt{43} }{ 2 }\)

The \(i\) doesn't go under the square root.

Answer 17

Oh. Thanks for helping me. :) Really appreciate it. My goal was to find all the complex zeros, but I still have no idea what the zeros are after doing all this work, lol. This is a difficult problem for me. I'm sorry, lol.

Answer 18

additional... the zeros you are referring to are the roots of the functions, the first root you got is a real number, and the last two are of complex numbers having an "i" term known as imaginary numbers...

Answer 19

I meant to say earlier that the quadratic formula would be used to find the complex zeroes of the quadratic function.

Answer 20

You found them

Answer 21

So the real number is 3 and the complex is \[\frac{ 7\pm i \sqrt{43} }{ 2 }\] Am I right?

Answer 22

Alright! Thank you so much @Hero and @Orion1213 for helping me! You guys are awesome!

Answer 23

Actually, I only covered with you the complex part of it.

Answer 24

There's a way to figure out if 3 is a zero.

Answer 25

Wasn't that done with the synthetic division though?

Answer 26

Your original function is

f(x)=x^3-10x^2+44x-69

If f(3) = 0, then 3 is a zero. If f(-3) = 0, then -3 is a zero.

It's one of the two.

Answer 27

Also, you can just graph the function to figure it out quickly

Answer 28

How do you graph it? I don't have a graphing calculator on me.

Answer 29

https://www.desmos.com/calculator/obtiejzym0

Answer 30

Oh you've already written the problem. That's the graph?

Answer 31

Now everything is finished right?

Answer 32

Yes, you found all the complex zeroes. By the way, 3 is also a complex zero.

3 + 0i

So include 3 in your solution of complex numbers.

Answer 33

So when I write my answers down it will look like this? Complex: \[\pm3\] and \[\frac{ 7pmi \sqrt{43} }{ 2 }\]

Answer 34

woops messed up on writing the last one. You get it, lol.

Answer 35

-3 is not a zero.

Answer 36

Oh thanks.