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So when we have something that is made up of two different terms |
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| that have been multiplied times themselves three times, | |||
| we have the difference of two cubes. | |||
| So X 3 - 1 factors like this ... | |||
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X3 - 1 = (X - 1)(X2 + X + 1) |
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| In general, if you have a 3X 3 - b 3, it factors to | |||
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(a3X3 - b3) = (a - b)(a2X2 + abX + b2) |
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| Examples: | |||
| X3 - 8 = | (X - 2)(X2 + 2X + 4) 8 = 23 | ||
| 27X3 - 1 = | (3X - 1)(9X2 + 3X + 1) 27 = 33 | ||
| Like with the squares, the numbers | |||
| don't have to be cubes of integers. | |||
| We can even write X - 1 | |||
| as the difference of two cubes. | |||
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| We couldn't factor X 2 + 1, can we factor X 3 + 1? | |||
| Yes we can!!! | |||
| It works out almost like the difference of two cubes. | |||
| The only changes are that the sign in the first factor is positive | |||
| and the middle sign in the second factor is a negative. | |||
| So: | |||
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X3 + 1 = (X + 1)(X2 - X + 1) |
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| Using our friends "a" and "b." | |||
| We can say ... | |||
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(a3X3 + b3) = (a2X2 - abX + b2) |
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| This one is called factoring the sum of two cubes. | |||
| Study these two different factoring plans for a moment or two. | |||
| The only difference is where the negative sign goes ... | |||
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(a3X3 - b3) = (a2X2 + abX + b2) |
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(a3X3 + b3) = (a2X2 - abX + b2) |
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copyright 2005 Bruce Kirkpatrick |
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