Question

Please help find all the zeros of the function

Answer 1

Where's the function, Dan?

Answer 2

×^4 -4×^3 +×^2 +16×-20

Answer 3

Sorry kind of slow in my phone @mathmale

Answer 4

So you have a fourth order polynomial: ×^4 -4×^3 +×^2 +16×-20, and need to find the four roots / solutions / zeros.

How have you factored polynomials (such as this one) in the past?

Answer 5

Well I know it's gonna have four answers because of the four roots but that's all I know, usually it has a way of simplifying to make it a quadratic or a sum of cubes or something like that

Answer 6

But that is all I know

Answer 7

This is a special kind of quartic. Hint:
\[x^4-4x^3+x^2+16x-20=(x^4-4x^3+5x^2)+(-4x^2+16x-20)\]

Answer 8

I'm sorry I'm still lost :(

Answer 9

Daniel: are you familiar with synthetic division? If not, are you familiar with long division?

Answer 10

Yes I know synthetic division

Answer 11

Cool. Now, Dan, look at the final term; it is -20. What are several possible factors of -20? We will test them (to see whether or not they are roots of the poly) through synth. div.

Answer 12

I tried 2 which is the first one I thought about

Answer 13

It works

Answer 14

What happened? Either there is a remainder or there is no remainder. You tried x=2 and found that there is no remainder after synth. div. Is that correct?

Answer 15

Yes sir there is no reminder

Answer 16

that's cool! So, you have already identified one root of your polynomial.

Dan, what are the coefficients of the resulting 3rd order polynomial?
You already have them...just go back to the work you did.

There should be four such coefficients.

Answer 17

1, -2, -3, 10

Answer 18

Cool! Now, Dan, what are some of the possible factors of 10?

Answer 19

1, 2, 5, 10

Answer 20

Try any of those as your newest synth. div. divisor.

Answer 21

2 doesn't work

Answer 22

5 neither

Answer 23

Just try another one then. try a negative one.

Answer 24

Oh, -2 it is

Answer 25

Yes. What are your remaining coeff?

Answer 26

1,-4 and 5

Answer 27

Yes, and what are factors, pos and neg, of that last coeff., 5?

Answer 28

1 and 5 only

Answer 29

But what about -1 and -5?

Answer 30

Those too

Answer 31

1 doesn't work

Answer 32

Use the quadratic formula on x^2 - 4x + 5 and see what roots result.

Answer 33

Oh forgot about the quadratic

Answer 34

How are you doing? Applying the Q. Formula, I've found two complex roots, and have been able to check one of them out (it's correct).

Answer 35

I got 2+-i

Answer 36

Same here. So, D., you have found all four roots of the given 4th order polynomial. Nice work!!

Answer 37

Any questions about what we've done here?

Answer 38

Thank you so much man, once again you are the best, I have no more questions for you