How do I factor x^7+2x^6-2x^5-4x^4+x^3+2x^2 ?

Question

agent0smith · Answer

$$\large x^7+2x^6-2x^5-4x^4+x^3+2x^2 $$
First notice you can pull out a common factor of x^2:
$$\large x^2(x^5+2x^4-2x^3-4x^2+x+2)$$
Now we want to find one factor of that, so we can use synthetic division. 

Hint: try checking x=1.

anonymous · Answer

Oh, yes, I did that, but I needed to find the roots as well. I saw on Wolfram Alpha, that it's supposed to end up looking like $$x^2(x-1)(x-1)(x+1)(x+1)(x+2)=0$$, but it doesn't explain exactly how to get that particular answer... Hmmm...

agent0smith · Answer

Do you know how to do synthetic division?

agent0smith · Answer

Dividing out the factor (x - 1) by synthetic division yields:
$$x^2(x-1)(x^4+3x^3+x^2-3x-2)$$
Notice if you put x=1 into the parentheses, you'll get zero again. So divide again by x=1, to pull out another (x-1) factor.

agent0smith · Answer

If you don't know how to do synthetic division, you might want to google it to see how it's done. Once you get it, it's easy.

anonymous · Answer

I've done synthetic division before. I'm taking a Calculus class right now. I just don't see why it wasn't pointed out in my previous reference. I'll study your answer. Thank You for the help!

agent0smith · Answer

Just keep dividing out factors until you're left with a quadratic, which you can then factor easily. You can use a couple of methods to find factors eg rational roots theorem.

anonymous · Answer

Okay. I see where it's headed. The (x-1) threw me off. Albeit, I'm still confused. How did that monomial get selected in the first place, from this whole equation? $$x^7+2x^6-2x^5-4x^2+x+1$$?

anonymous · Answer

Whoops. Correction to the previous equation. It should look like this instead: $$x^7+2x^6-2x^5-4x^4+x^3+2x^2=0$$.

agent0smith · Answer

I chose x=1 because if you put x=1 into that equation, it equals zero... so (x-1) has to be a factor. Same thing with the next x=1 factor.

anonymous · Answer

Okay. I'll take note of that. Thanks again!

agent0smith · Answer

(x-1) is often a factor in these types of questions, so it's good to check x=1 in the equation first.

anonymous · Answer

Ah, I see...

anonymous · Answer

nevermind idiots.

agent0smith · Answer

k.

anonymous · Answer

@Fireflame13: You spelled "nevermind" wrong.

anonymous · Answer

did i spell ugly wrong?