| Now we actually get to use this factoring for something! | ||
| If we have something like ... | ||
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| Can we make this easier to deal with? | ||
| That is, can we make the fraction look simpler? | ||
| As it sits, no we can't. | ||
| But if we factor it ... | ||
| Let's see: | ||
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| We can write this as: | ||
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| When we have a fraction where all the things are multiplied | ||
| like factors, we can split up these factors into different fractions. | ||
| Using a's and b's, we might say ... | ||
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| When we have stuff like this, | ||
| the factors don't care how they are split ... | ||
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| Let's see what damage we can do to our problem | ||
| with this trick ... | ||
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| Now look at those puppies we split apart. | ||
| One has the same term in the numerator and denominator. | ||
| That means it is equal to 1. | ||
| So we can say ... | ||
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| Multiplying something times 1 doesn't change the value | ||
| so we don't even need to write it in if we don't want to. | ||
| What we do in real life when we have the same thing | ||
| on the top and the bottom of a fraction is to cancel. | ||
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| Remember that to do this, | ||
| everything in the numerator and denominator | ||
| must be multiplied together. | ||
| If we have: | ||
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| We can not do any canceling because of that 2. | ||
| Let's see another one: | ||
| Example: | ||
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| Factoring this we get ... | ||
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| Then canceling ... | ||
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| OK. How about this one ... | ||
| Example: | ||
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| The numerator is not a perfect square, | ||
| but it can be factored. | ||
| The denominator is the difference of 2 squares: | ||
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| Example: | ||
| What would we do with something like ... | ||
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| What we do is this ... | ||
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| Now before you start multiplying these things out | ||
| try to factor then all to see if there are any factors | ||
| that you can cancel. | ||
| If you work with it for a while, | ||
| you will find that all 4 of these terms will factor. | ||
| Eventually you get: | ||
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| Now since everything is multiplied by everything else | ||
| on the top and on the bottom, | ||
| we can cancel any time we find the same term | ||
| on the top and on the bottom. | ||
| Is there anything to cancel? | ||
| Yup. | ||
| In fact, everything cancels! | ||
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| Hey, if EVERYTHING CANCELS, what's left? | ||
| Remember that before, when we started this canceling business, | ||
| we said that we were going to get rid of things | ||
| that were equal to one. | ||
| What we actually did here is: | ||
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| So as weird as it looks ... | ||
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| Example: | ||
| What would you do with this one? | ||
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| There is all kinds of fancy math hocus pocus to prove it, | ||
| but the deal is this ... | ||
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| So you flip the thing you were dividing by | ||
| and then multiply ... | ||
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| Now let's solve it! | ||
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| So we get ... | ||
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| Can we factor this any farther? | ||
| Sure. | ||
| We could factor the (X - 3) as the difference of two squares. | ||
| It would be weird, but we know how. | ||
| We could factor the (X + 2) as the sum of two cubes. | ||
| That's even more weird. | ||
| The question is, why would we do it? | ||
| We can't cancel anything else | ||
| so there's no reason to do any more factoring. | ||
| Should we multiply it out? | ||
| If you have a reason to, sure. | ||
| If not, why bother. | ||
| I can't think of a reason to, can you? | ||
|
copyright 2005 Bruce Kirkpatrick |
