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There are some factors that show up all the time. |
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| It is a good idea to get to know these. | |||
| They are ... | |||
| The Perfect Squares: | |||
| Using "a" and "b" to stand for any old numbers we might have, | |||
| the perfect squares are: | |||
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(aX + b)2 = a2X2 + 2abX + b2 |
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(aX - b)2 = a2X2 - 2abX + b2 |
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| When these two are multiplied out like this | |||
| they are almost the same. | |||
| The only difference is that the one that starts with a minus | |||
| has a minus on the middle term. | |||
| So using real numbers and multiplying these out | |||
| we would get ... | |||
| Examples: | |||
| (X + 3)2 = 12X2 + 2x1x3xX + 32 = X2 + 6X + 9 | |||
| (3X + 5)2 = 32X2 + 2x3x5xX + 52 = 9X2 + 30X + 25 | |||
| (X2 - .5)2 = .52X4 - 2x1x.5xX2 + .52 = .25X4 - X2 + .25 | |||
| The Perfect Cubes: | |||
| Using "a" and "b" to stand for any old numbers we might have, | |||
| the perfect cubes are: | |||
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(aX + b)3 = a3X3 + 3a2bX2 + 3ab2X + b3 |
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(aX + b)3 = a3X3 - 3a2bX2 + 3ab2X - b3 |
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| When these two are multiplied out, | |||
| the only difference between them is also in the signs. | |||
| The signs on (aX + b) 3 are all positive | |||
| The signs on (aX - b) 3 alternate: positive - negative - positive - negative | |||
| So using real numbers and multiplying these out | |||
| we would get ... | |||
| Examples: | |||
| (X + 3)3 = 13X3 + 3x12x3xX2 + 3x1x32xX + 33 = X3 + 9X2 + 27X + 9 | |||
| (X - p)3 = 13X3 - 3x12xpxX2 + 3x1xp2xX - p3 = X3 + 3pX2 + p2X + p3 | |||
| We could keep on going with this stuff forever. | |||
| We could do (aX + b) 4, (aX - b)7 and on and on, | |||
| but there is an easier way ... | |||
| It's called Pascal's triangle. | |||
| It starts like this. | |||
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1 1 |
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| For our purposes here, | |||
| these stand for the two terms in something you | |||
| would be multiplying times itself. | |||
| Since this is the start, | |||
| it shouldn't be too surprising that this stands for: | |||
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(a + b)1 which is just a + b |
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| The next step builds on the start above (the 1 1). | |||
| It looks like this ... | |||
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| The following levels continue the pattern. | |||
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| OK, this looks really cute. | |||
| BUT WHAT GOOD IS IT? | |||
| Here's the deal ... | |||
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| See what happens? | |||
| The coefficients are the numbers from the triangle. | |||
| Check out what the exponents on X are doing. | |||
| It's a countdown ... | |||
| We could have a coefficient on X. | |||
| Or a number other than 1. | |||
| To deal with that, | |||
| here's the first 4 levels of the triangle | |||
| with "a" and "b" standing for any old numbers we might have. | |||
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| Look what happens. | |||
| The numbers in bold are the Pascal's triangle numbers. | |||
| The exponents on X go down 1 every term as we go from left to right. | |||
| The exponents on "a" go down 1 every term as we go from left to right. | |||
| The exponents on "b" go up 1 every term as we go from left to right. | |||
| This stuff works real well, but it takes up a lot of space. | |||
| If I wanted to write out a Pascal's triangle | |||
| for exponents up to, say, 50, | |||
| It would take up an amazing amount of space. | |||
| Math types try to save space anywhere they can. | |||
| They came up with a shorthand way to tell someone | |||
| what any level of the triangle would look like. | |||
| It goes like this ... | |||
| Using "a" "b" and "n" to stand for | |||
| any coefficient or exponent we might have ... | |||
| (aX + b)n = anXn - nan-1bXn-1 ... nabn-1X + bn | |||
| This is the start of something called the binomial theorem. | |||
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copyright 2005 Bruce Kirkpatrick |
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