Why does the negative stay with the fraction after factoring out a -1 when dealing with opposite factors?

Question

So, I understand what I'm supposed to do when coming across opposite factors when simplifying rational expressions. For example:

$\dfrac{4-w}{w^2-8w+16}$ simplifies to $\dfrac{4-w}{(w-4)(w-4)}$

So I know you're supposed to factor out the negative from $4-w$, and when you do that you get $-1(4-w)$ which equals $(w-4)$.

So, the $w-4$ cancels out and you're left with one $\dfrac{1}{w-4}$, but the negative is still there, so it would be $\dfrac{-1}{w-4}$.

I don't get why the negative is still there, though, since I distributed it to the $4-w$ to get $w-4$. Can someone explain why I'm supposed to keep the negative sign?

I'm really sorry my writing is kind of confusing, I have been trying to figure out how my make fractions on this website and I still can get it to work.

Answer 1

Either way,$$\begin{align}\frac{4-w}{w^2-8w+16}&=\color{red}{-\frac{w-4}{(w-4)^2}}=\color{blue}{\frac{4-w}{(4-w)^2}}\\&=\color{red}{-\frac1{w-4}}=\color{blue}{\frac1{4-w}}\end{align}$$ for $w\neq4$.

Answer 2

we have $$\frac{4-w}{(w-4)(w-4)}=\frac{-(w-4)}{(w-4)(w-4)}$$ if $w\ne 4$

Answer 3

So I know you're supposed to factor out the negative from $4-w$, and when you do that you get $\color{red}{(-1)(4-w)}$ which equals $(w-4)$

What you've done here is not “factored out the negative”; you've just multiplied by $-1$. Factoring an expression creates an equal expression, while multiplying by $-1$ doesn't.

In order to factor out the negative, you need to write $$ 4-w = (-1)(-4+w) = (-1)(w-4) $$ Then, after cancellation, the $-1$ remains.