Question

Find all zeros by factoring each function. f(x)= x^4 - x^2 - 30. Explain in detail please.

Answer 1

Substitute y=x^2,
so
f(y)=y^2-y-30
After that, find x = sqrt(y) (note x could be complex)

Answer 2

Complex?

Answer 3

imaginary, when we take sqrt(-1), the result is i.
Suppose we obtained y=4 or y=-3.
Then
\( x=\pm sqrt(4)\ or\ x=\pm sqrt(-3) \)
which translates to
\( x=-2,\ 2,\ -sqrt(3)i,\ or\ sqrt(3)i \)

Answer 4

The final answer is sqrt(6), -sqrt(6), sqrt(5)i, -sqrt(5)i. but im trying to find out the steps my teacher took to find the answer.

Answer 5

Start by factoring:
g(y)=y^2-y-30

Answer 6

Ok, I have that part, its (x^2 - 6)(x^2 + 5). Then i set the x's to zero, right? that's where i'm lost.

Answer 7

Yes, y=x^2=6 => x^2=6, which can be solved as \( x=\pm\sqrt{6} \)
You will similarly calculate the value of x for the other solution y=-5. (result is complex, see example above)

Answer 8

Sorry, \( y=x^2-6=0 => x^2=6 => x=\pm\sqrt{6} \)

Answer 9

So I factor again? Then is your complex answer an example or...?

Answer 10

Similarly,

\( y=x^2-(-5)=0 => x^2=-5 => x=\pm\sqrt{-5}=\pm \sqrt{5} i \)

where \( i = \sqrt{-1} \)

Answer 11

Ok, so that is the factored function? So => is a symbol for?

Answer 12

=> means "implies", or "therefore"

Answer 13

So if i wanted to write this on my paper i would write it as...

Answer 14

But i'm still confused. After you factor the 1st part, you then factor what you factored?

Answer 15

You can also write them on separate lines, one after another, as long as each line follows from the previous.

Answer 16

Such as
\( x^2=16\)
\( x=\pm 2 \)

Answer 17

Oh, Ok. Makes sense. But your answer is a little complicated. Can you break it down more please?

Answer 18

where y+5=0?

Answer 19

Yes, that problem was complicated.

Answer 20

Sorry for being to condensed!
\( (y+5)=0 \)
\( (y=-5 \)
\( x^2=-5 \)
\( x=\pm \sqrt{-5} \)
\( x=\pm \sqrt{5} \ i\)
Do you follow the steps better now?

Answer 21

If you don't, do let me know. It's no use to copy the step, get 100% for the homework, and get 10% for the exam!

Answer 22

Alright, so the step you just put is after i factor the first time?

Answer 23

Sorry, all the x's should have been y's. Deleting the previous post to avoid confusion.
Yes. After the quadratic in y had been factored, there were two cases, either (y-6)=0, or (y+5)=0
This is for the second case, y+5=0.
I believe you did the (y-6)=0 case.

Answer 24

Yes, the (x-6) was what my teacher did. So let me break this down from the beginning, i factor the first part, then i factor again to solve for y? Then i get my initial answer?

Answer 25

The final answer should consist of 4 values (for a quartic).
Two of these correspond to (y-6)=0, and the other two from (y+5)=0.

Answer 26

And your work above explains your answer right?

Answer 27

Yes, you will need to combine the work for the two cases.

Answer 28

Ok, Got you. So (x+5) = 0?

Answer 29

Then x=-5, then x^2=-5, x= +- sqrt-5, x= +-sqrt5i? and then the same for x^2 -6=0?

Answer 30

yep!