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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Which of Wendy Jones' books are these questions coming from by the way? (I'm referring to your post in this thread, in which you mention the self assessments in one of her books.)

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