Question

Determine the roots (zeros) of x^4-3x^3+5x^2-27x-36 = 0
i got one of the zeros from the calculator and it was -1
from that i used synthetic division to get my new equation : x^3-4x^2+9x-36. now i dont know how to get the rest of my zeros. i need help!

Answer 1

hint: factor out x^2 from the first two terms, and 9 from the second two terms.

Answer 2

hmmm have you covered the "rational root test" yet?

Answer 3

or ... are you supposed to get them from their graph?

Answer 4

no, my test is tomorrow. this is like a review packet and that is the exact question. @jdoe0001 and @ertdre i did i got (x^2+9) (x-4)
that would give me one of my other answers being 4, right? i dont know where to go from there.

Answer 5

so... how are you supposed to find the roots then? what are you meant to use to get them?

Answer 6

or what are you expected to use to get them ?

Answer 7

@jdoe0001 have you not unlocked human calculator at least??

Answer 8

well im supposed to use synthetic division to get one of the zeros and from the new equation i am supposed to get the rest of my zeros

Answer 9

How did you get that polynomial when you assumed one of the zeros was x=-1

Answer 10

I'm not getting that polynomial (and I'm not getting a remainder 0)

Answer 11

which means x=-1 wasn't a zero

Answer 12

unless I have done some mistake

Answer 13

i got it by doing synthetic division. @myininaya

Answer 14

-1| 4 -3 5 -27 -36
| -4 7 -12 39
-------------------------
4 -7 12 -39 | 3
not remainder 0 so x=-1 is not a zero

Answer 15

you forgot the 1

Answer 16

Oh the coefficent of x^4 is 1 not 4

Answer 17

yes maam the 4 is the degree

Answer 18

-1| 1 -3 5 -27 -36
| -1 4 - 9 36
- --------------------
1 -4 9 -36 |0
okay x=-1 does work

Answer 19

yeah i thought there was a 4 in front also

Answer 20

Well you already found the other zeros by factoring that resulting polynomial

Answer 21

\[x^3-4x^2+9x-36=0 \\x^2(x-4)+9(x-4)=0 \\ (x-4)(x^2+9)=0 \\ x-4=0 \text{ when } x=?_1 \\ x^2+9=0 \text{ when } x=\pm ?_2\]

Answer 22

i get everything untill the last line. i dont know how you did that

Answer 23

how I got x^2+9?

Answer 24

I didn't get that
You actually did

Answer 25

well and then I also got it after I factored
but you got x^2+9 as a factor first

Answer 26

Do you mean do hot you solve x^2+9=0 for x?

Answer 27

i mean your answer to x^2+9

Answer 28

\[x^2+9=0 \\ x^2=-9 \]
Now you just take the square root of both sides

Answer 29

i am officially confused

Answer 30

would it be 3i

Answer 31

you would have two solutions

Answer 32

+ or - 3i

Answer 33

heheh officially =) so you're "unofficially" doing ok then ? =)

Answer 34

okay i got it thank you! @myininaya & hah nope now i am officially doing okay because i got exactly what i needed. @jdoe0001

Answer 35

so the part that was confusing was just solving the x^2+9=0 thing?

Answer 36

Or was it why I set x^2+9=0

Answer 37

Or was it where x^2+9 came from?

Answer 38

it was the solving x^2+9=0 part. i thoight that i was supposed to factor it. i had everything exept that part. i was trying to factor but i saw no way to factor so i was confused

Answer 39

\[x^4-4x^3+5x^2-27x-36=0 \\ (x+1)(x-4)(x^2+9)=0\]

Answer 40

you can factor x^2+9

Answer 41

\[(x-3i)(x+3i)\]

Answer 42

\[x^2+a^2 =(x-ai)(x+ai)\]

Answer 43

i never thought about that though.

Answer 44

\[(x-ai)(x+ai)=x^2+xai-aix-a^2i^2 \\ =x^2+0-a^2(-1) \\ =x^2-a^2(-1) \\=x^2+a^2\]

Answer 45

You don't have to factor x^2+9 to find the zeros of x^2+9 though

Answer 46

yeah i know that now.

Answer 47

\[x^2+a^2=0 \\ x^2=-a^2 \\ \sqrt{x^2}=\sqrt{-a^2} \\ |x|=\sqrt{-a^2} \\ x=\pm \sqrt{-a^2} \\ x= \pm i \sqrt{a^2} \\x=\pm ai\]