| There is a shortcut that we can use | ||
| to see if something like (X - 2) is a factor | ||
| of some big polynomial like 2X 6 - 4X 5 + 3X 3 - 11X 2 + 17X - 14 | ||
| To use this shortcut, the factor that we're checking | ||
| must start with X, not something like X 2 or 4X. | ||
| Example: | ||
| Check to see if (X - 2) is a factor of | ||
|
2X6 - 4X5 + 3X3 - 11X2 + 17X - 14 |
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| Here's what we do ... | ||
| Step 1) | ||
| Make sure that there is a term | ||
| for every power of X less than the biggest one. | ||
| If any are missing (like X4 here) write a zero in it's place ... | ||
|
2X6 - 4X5 + 0 + 3X3 - 11X2 + 17X - 14 |
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| Step 2) | ||
| All powers of X are now present? Good! | ||
| Now write down just the coefficients | ||
| (the numbers without the X's or exponents). | ||
|
2 - 4 + 0 + 3 - 11 + 17 - 14 |
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| Step 3) | ||
| Now look at the term that we want to check. | ||
| Look at it as: | ||
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| we need whatever goes in the box. | ||
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Since our term is (X - 2), we have a 2 in the box. |
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| BUT if we were checking (X + 5) | ||
| we would need to write it as (X - (-5)). | ||
| That means that -5 would be in the box. | ||
| Step 4) | ||
| Take whatever goes in the box AND | ||
| the coefficients from step 2 and write them all down. | ||
| Then draw a line under the numbers ... | ||
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| Step 5) | ||
| Take the first coefficient (here it's a 2) | ||
| and write it below the line ... | ||
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| Step 6) | ||
| Multiply the number that you wrote under the line | ||
| times the number in the box. | ||
| Write the answer under the next number to the right (2 x 2 = 4) ... | ||
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| Step 7) | ||
| Add the number you just wrote down | ||
| and the number above it. | ||
| Write the answer below the line (-4 + 4 = 0) ... | ||
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| Step 8) | ||
| Repeat steps 6 and 7 until you run out of numbers on the right ... | ||
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| Step 9) | ||
| IF the last number that you write down under the line is a zero, | ||
| the thing that you were checking IS a factor of the big polynomial. | ||
| If the last number is NOT a zero, | ||
| the thing that you were checking is not a factor. | ||
| Step 10) | ||
| If the thing that you were checking | ||
| is a factor of the big polynomial, | ||
| the other factor is built from the numbers under the line. | ||
| We write powers of X next to the numbers. | ||
| Start on the left with an exponent one smaller | ||
| than the biggest exponent in the big polynomial. | ||
| As we go to the right, the exponents get one smaller each number. | ||
| That is ... | ||
|
2X5 + 0X4 + 0X3 + 3X2 - 5X + 7 |
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| We can leave out any term with a zero coefficient. | ||
|
2X5 + 3X2 - 5X + 7 |
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| That means ... | ||
|
2X6 - 4X5 + 3X3 - 11X2 + 17X - 14 = (X - 2)(2X5 + 3X2 - 5X + 7) |
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| That's just swell. | ||
| Now we've got good news, bad news, and good news. | ||
| Good News: Once you get the hang of it, | ||
| this is much easier to do than equation long division. | ||
| Bad News: This trick only works with X ± (a number) | ||
| and we still need to guess at what the factors might be. | ||
| Good News: This little trick has another tricky use! | ||
| Say that we wanted to find the value of: | ||
|
2X6 - 6X5 - 8X4 - X2 + 6X - 10 when X = 4 |
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| Up to now, what we would have to do is calculate 4 6, | ||
| then multiply that by 2, | ||
| then calculate 4 5, | ||
| then multiply that by -6, | ||
| add that to the first calculation, and so on. | ||
| It's a big mess and takes forever! | ||
| BUT NOW, we have a new way to figure this out. | ||
| Here's what we do ... | ||
| Set up the polynomial coefficients | ||
| just like we did in the example. | ||
| Fill in any missing powers of X with zeros ... | ||
|
2 -6 -8 0 -1 6 -10 |
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| The X value that we want to solve for | ||
| goes in the box on the left ... | ||
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| Draw the line and do the trick ... | ||
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| Here's what makes this so great. | ||
| The last number that you wrote down (the -2) | ||
| is the value of 2X 6 - 6X 5 - 8X 4 - X 2 + 6X - 10 when X = 4. | ||
| It's true! It works! | ||
| Don't believe me? | ||
| Check it out the old way ... | ||
|
copyright 2005 Bruce Kirkpatrick |
