Question

Find all zeros. Show synthetic division or factor.
f(x) = x^4 + 2x^3 + x^2 - 4

Answer 1

@jim_thompson5910 ?

Answer 2

First list all the possible factors of 4, what are they?

Answer 3

1,2,4?

Answer 4

-1,-2,-4 is also included as well

Answer 5

so those are your possible rational roots

Answer 6

But are those the zeros? How would it be shown in synthetic division?

Answer 7

say we want to test x = -1 as a root (ie see if it's a root)

then this is what we get for our synthetic division table


-1 | 1 2 1 0 -4
| -1 -1 0 0
--------------------------------------------------
1 1 0 0 -4

Notice how the last value in the last row is -4

This means that x = -1 is NOT a root

Answer 8

Okay makes sense. And a root is refered to as a zero correct?
What WOULD be the roots?

Answer 9

You would repeat this table, but do it for x = 1, x = 2, x = 4, etc for all the possible roots

Answer 10

if you get 0 as the last value in the last row, then that value is a root or zero

Answer 11

and yes, root = zero of a function

Answer 12

Okay let me test.

Answer 13

It's not -2 correct?

Answer 14

what value did you get in the last row

Answer 15

-8 if I did it right.

Answer 16

you didn't, you went wrong somewhere

Answer 17

Let me try again.

Answer 18

alright

Answer 19

Oh okay found my mistake. It is -2. What's the next step?

Answer 20

you should have gotten this

-2 | 1 2 1 0 -4
| -2 0 -2 4
--------------------------------------------------
1 0 1 -2 0

did you get this?

Answer 21

Yes I did.

Answer 22

ok so the last row 1 0 1 -2 0

translates to

x^3 + 0x^2 + 1x - 2 remainder 0

which turns into

x^3 + x - 2

Answer 23

now you repeat the whole process we did above to find the possible roots and check them

Answer 24

to save time, the possible roots of x^3 + x - 2 are: -1, 1, -2, 2

Answer 25

possible rational roots*

Answer 26

Thanks! So with the x^3 + x - 2 equation I redo the synthetic division process? Like this: What am I missing?

Answer 27

it should be

-2 | 1 0 1 -2
| -2
----------------------------------------
1

I'll let you finish the table

Answer 28

I think I messed up.

Answer 29

you did it right, i got this

-2 | 1 0 1 -2
| -2 4 -10
----------------------------------------
1 -2 5 -12

which is exactly what you have

Answer 30

so -2 is NOT a root to x^3 + x - 2

Answer 31

Oh okay. :)

Answer 32

So do I test 2, 1, -1 now?

Answer 33

yes

Answer 34

Okay so the correct root is 1.

Answer 35

good

Answer 36

what table do you get for x = 1

Answer 37

Next?

Answer 38

when you did synthetic division, what table did you get for x = 1