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Question

I can't believe that I seriously ask this question as it is so simple.

Given this

$-x^3+4x$

I'd like to factor out -x, so I did

$-x(x^2-4)$

which equals

$-x(x^2-2^2)$

equals

$-x(x-2)(x+2)$

Right?

However, seems like the guys in the lecture script I am reading factored out "x" instead of -x. They have

$x(2-x)(2+x)$

I thought I could multiply my result with -1 to get the same. But that gives me

$x(2-x)(-2-x)$

But thats wrong. Wtf am I doing wrong? I am struggling with the simplest thing! :(

Answer 1

$$x(2-x)(2+x)=x(-1)(x-2)(2+x)=-x(x-2)(x+2)$$

Since the two quantity is the same, you should not expect that multiplying $-1$ to the lecture script answer should give you your answer.

Answer 2

$$-x(x-2) = +x(2-x) $$ so your answer matches the lecture answer.

Answer 3

You multiplied every factor by $-1$ which means you multiplied by $-1\cdot-1\cdot-1 = -1$ which indeed will change the result. Instead you should multiply by a special form of $1$ (which means multiplying an even number of factors by $-1$). So to match the lecturer's factorization you would do:

$$ -1\cdot (-x) = x \\ -(x-2) = 2-x $$

Then you, correctly get:

$$ x(2-x)(x+2) = x(2-x)(2+x) $$

Answer 4

if your operation is in the ring of caracteristique 2, there are no error because $-1=1$, but if the caracteristic is not 2, thene $1\not=-1$ and you have macked one error in the calcul that must $-x(x-2)(2+x)=(-1)(x)(x-2)(2+x)=(-1)(2+x)(x)(x-2)=(-2-x)(x)(x-2)=x(x-2)(-2-x)$