MHB Did I misinterpret the sign attached to a term when factoring?

AI Thread Summary
The discussion revolves around the correct factoring of the expression xy+x-2y-2. The original attempt resulted in x(y+1)-2(y-1), which was incorrect due to a misunderstanding of sign when factoring out -2. The correct factorization is x(y+1)-2(y+1), leading to (x-2)(y+1). The key takeaway is that when factoring out a negative, the signs within the parentheses must change, highlighting the importance of careful sign management in algebraic manipulation. Understanding these nuances is crucial for accurate factoring in future problems.
find_the_fun
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I wanted to factor [math]xy+x-2y-2[/math]
I got [math]x(y+1)-2(y-1)[/math] and got stuck

I tried somethings out and noticed [math](x-2)(y+1)=xy+x-2y-2[/math] so how come I did get stuck? Did I extract the [math]2[/math] incorrectly with the sign? For example should I have interpreted it at [math]-2y-2[/math] not [math]2y-2[/math]?
 
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find_the_fun said:
I wanted to factor [math]xy+x-2y-2[/math]
I got [math]x(y+1)-2(y-1)[/math] and got stuck

I tried somethings out and noticed [math](x-2)(y+1)=xy+x-2y-2[/math] so how come I did get stuck? Did I extract the [math]2[/math] incorrectly with the sign? For example should I have interpreted it at [math]-2y-2[/math] not [math]2y-2[/math]?

Yes, you simply factored incorrectly.

[math]xy+x-2y-2=x(y+1)-2(y+1)=(x-2)(y+1)[/math]

When you factor a negative factor from a group of terms, the signs of everything within the resulting parentheses must change.
 
MarkFL said:
Yes, you simply factored incorrectly.

[math]xy+x-2y-2=x(y+1)-2(y+1)=(x-2)(y+1)[/math]

When you factor a negative factor from a group of terms, the signs of everything within the resulting parentheses must change.

But [math]2y-2[/math] can be factored to [math]2(y-1)[/math]. Since [math]2y-2[/math] appears in the equation can't we say [math]...-(2y-2)=...-(2(y-1))[/math]?
 
It is $-2y-2$ that appears in the expression...and both terms have $-2$ as a factor. :D
 
find_the_fun said:
But [math]2y-2[/math] can be factored to [math]2(y-1)[/math]. Since [math]2y-2[/math] appears in the equation can't we say [math]...-(2y-2)=...-(2(y-1))[/math]?

That's the point. [math]2y-2[/math] does not appear in the equation.
You can read the equation as:
$$xy+x−2y−2 = xy+x+(−2)y+(−2)$$

As you can see $(−2)y+(−2)$ is definitely different from $2y-2$.

From here we can get to:
$$xy+x+(−2)y+(−2) = x(y+1)+(−2)(y+1)$$

Which can also be written as:
$$x(y+1)−2(y+1)$$
 
I guess I see what I did wrong, I did something like, took a -2 inserted a bracket and pretend it was positive. I try to understand exactly what went wrong in hopes of not making the same mistake again.
 
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