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Fl4sh

Simple question

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Factor x(x – 2) + 3(2 – x).
This is almost the same as the previous case, but not quite, because "x – 2" is not quite the same as "2 – x". If I'd had "x + 2" and "2 + x", they factors would have been the same, because order doesn't matter for addition. But order does matter for subtraction, so I don't actually have a common factor here.

But I would have a common factor if I could just flip (or "reverse the order of") that subtraction. What would happen if I did that? Take a look at the following numerical subtraction:

5 – 3 = 2

3 – 5 = –2

When I flipped the subtraction in the second line, I got the same answer except that the sign had changed. This is always true: When you flip a subtraction, you also change the sign out front. In our case, this means:

x(x – 2) + 3(2 – x) = x(x – 2) – 3(x – 2)

By reversing the subtraction in the second parenthetical, I have created a common factor, so I can now proceed as I had in the previous example:

x(x – 2) + 3(2 – x) = x(x – 2) – 3(x – 2)

= (x – 2)(x – 3)

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How come you can just flip the 3's sign to negative. Someone explain this concept. ;o

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When transforming equations like that, you can always multiply by 1, so long as your definition of 1 does not contain your unknowns. Likewise, you can add 0 to fix things up.

So,
3(2-x) => (-1)(-1)(3)(2-x) => (-1)(3)(-1)(2-x) => -3(x-2)
Since (-1)*(-1) is 1, we didn't actually change anything. Its quite similar to how you reduce fractional addition.
1/2 + 1/3? 1/2 * 3/3 + 1/3 * 2/2 => 3/6 + 2/6
We multiply both parts by 1, in order to make the numbers easier to work with.

Likewise you could add 0, as I mentioned.
x2 + 2x => x2 + 2x + (1 - 1) => (x+1)(x+1) - 1

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