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If you have two factors like: |
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(X + 1)(X - 1) |
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| Look what happens when you multiply them out ... | |||
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(X + 1)(X - 1) = X2 - X + X - 1 = X2 - 1 |
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| The middle terms cancel each other out! | |||
| What we get at the end is called the difference of two squares. | |||
| They call it that because: | |||
| It is two terms. | |||
| The second one is subtracted. | |||
| They can both be gotten by multiplying something times itself. | |||
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X × X = X2 and 1 × 1 = 1 |
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| When we are factoring stuff, | |||
| we are going in the other direction. | |||
| We start with the multiplied out thing | |||
| and try to find the factors. | |||
| Say we have ... | |||
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X2 - 9 |
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| and want to factor it. | |||
| We have no middle term. | |||
| We just have an X 2 and a number. | |||
| The number is negative. | |||
| You can get the number by multiplying 3 times itself. | |||
| We can factor this as: | |||
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(X + 3)(X - 3) |
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| Here's a tougher one. | |||
| Factor this ... | |||
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4X2 - 25 |
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| We have no middle term. | |||
| The number is negative. | |||
| You can get the number by multiplying 3 times itself. | |||
| But instead of X 2, we have 4X 2. | |||
| That's not a problem ... | |||
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2X x 2X = 4X2 and 5 x 5 = 25 |
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| So we can factor this ... | |||
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4X2 - 25 = (2X + 5)(2X - 5) |
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| In general, the formula for doing these is: | |||
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| Here are a few more examples ... | |||
| 4X2 - 9 = | (2X + 3)(2X - 3) | ||
| 16X2 - 1 = | (4X + 1)(4X - 1) | ||
| X4 - 81 = | (X2 +9)(X2 - 9) = (X2 +9)(X + 3)(X - 3) | ||
| We got a little trickier on that last one! | |||
| If you have any big even power (like X 4), | |||
| you can just chop the power in half in the factors (make them X 2's). | |||
| If one of those factors is still the difference of two squares, | |||
| you can chop it in half again. | |||
| If the numbers in the problem | |||
| are not squares of whole numbers, | |||
| we can still do this stuff. | |||
| Here's some examples of that ... | |||
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| As long as we have a minus sign between the two terms, | |||
| we can use this trick to factor them. | |||
| Could we factor stuff like X 2 + 1? | |||
| For now, no. | |||
| Later on, you may learn about | |||
| something called IMAGINARY NUMBERS. | |||
| They can help to factor stuff like this. | |||
| But for now, the answer to: | |||
| "Can X 2 + 1 be factored?" is no. | |||
| OK here's another question. | |||
| All the powers we have talked about factoring so far are even. | |||
| What if we had an odd power? | |||
| Could we factor X 3 - 1? | |||
| Might this be called the difference of two cubes???? | |||
| Yup! It is. | |||
| And on the next page, you will learn all about it. | |||
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copyright 2005 Bruce Kirkpatrick |
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