Question

Find roots of the polynomial P(x) = x^3 + 4x^2 - 4x - 16

Answer 1

this polynomial is factorable by grouping; can you factor by grouping?

Answer 2

Not really. Factoring always has confused me

Answer 3

\[x^3+4x^2-4x-16=x^2(x+4)-1(x+4)\]

Answer 4

do you see how i did that? or no?

Answer 5

I really hate to look stupid, but I have no idea what you did to get that.

Answer 6

if you knew how to do it you wouldn't be here :) i'm here to help you so ask whatever you want ok? look at the first two terms, what factor do they have in common?

Answer 7

x^2?

Answer 8

yes! and i took out the x^2 and (x+4) is left that's where x^2(x+4) comes from. now the last two terms have -x - 4 the only thing common there is -1. So the first step factors into

\[x^2(x+4)-1(x+4)\]

Answer 9

now what do those two terms have in common?

Answer 10

Why do you have the z?

Answer 11

I don't see a z, you mean

\[x\]

Answer 12

Ha, yeah! What is that?? It's confusing.

Answer 13

thats an X

Answer 14

is this better?
\[X^2(X+4)-1(X+4)\]

Answer 15

MUCH

Answer 16

ok now what's in common?

Answer 17

x+4

Answer 18

i made a mistake because i copied the problem down wrong its minor but it should be

\[X^2(X+4)-4(X+4)\]

Answer 19

left a 4 off... but yes (X+4) is the common factor. When I've tutored this is usually the step people struggle with the most to factor out the (X+4) but when you do it looks like this:

\[(X+4)(X^2-4)\]

Answer 20

Do you see how i got that?

Answer 21

Yes. You took what they had in common and the outside of what they had in common. If that made sense. It made sense to me

Answer 22

good now is there anything else to factor in that?

Answer 23

Not that I see?

Answer 24

what about X^2-4

Answer 25

Oh yeah.
It would be (x-2)(x(-2)

Answer 26

Nice! so now we have:
\[f(X)=(X+4)(X+2)(X-2)\]
Can you tell what the zeroes are now?

Answer 27

-4, -2 and 2??

Answer 28

:) good job

Answer 29

So the terms "roots" and "zeros" mean the same thing?

Answer 30

yes

Answer 31

Oh! Thank you!

Answer 32

they have similar meaning. the roots make the polynomial zero

Answer 33

they get trickier when they can't be factored; luckily in this scenario wasn't the case

Answer 34

Okay, that makes sense now.

Answer 35

synthetic division is something i probably couldn't try to type here, that's another way of checking the roots

Answer 36

I for some stupid reason can do synthetic division in my head. Just can't factor to save my life

Answer 37

you're better than me i have to write it down. check with synthetic division -4, -2 and 2 into that polynomial see if you get 0

Answer 38

I'm an all or nothing kind of person. XD