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First, set up the long division. Although the dividend skips the terms x3 and x2, what do we do? That's right, we include 'em. Write them as 0x3 and 0x2 in case we need those columns later for subtraction. If it turns out we don't need them, oh well. We can always stick them in our scrapbook, look back at them years later, and smile.
Now we look to see how many times the first term of the divisor goes into the first term of the dividend, perform the appropriate subtraction, and bring down the remaining terms of the polynomial:
It looks like we did, in fact, need that x3 column, so it's a good thing we had it. Otherwise, we wouldn't have had anywhere to write -2x3 that would work so well for the subtraction. We would have been up polynomial creek without an exponent, so to speak.
Next we see how many times 3x2 goes into the first term of our new polynomial, subtract, and bring down terms:
By the way, we needed the x2 column too. Thank goodness we included him. Isn't it nice not having any regrets? Now we see how many times 3x2 goes into 6x3, and so on:
Finally, 3x2 goes into 6x2 twice:
The final answer is
Check the answer to the above example by multiplying (3x2 – 2) and (3x3 + 3x2 + 2x + 2). You should get 9x5 + 9x4 – 4x – 4. When you do, you should feel a sense of validation wash over you. On the other hand, it might just be raining.
We don't always need to write out the zeros. Sometimes there are so many terms with the coefficient zero that writing them all out can be confusing and do more harm than good. To avoid confusion, we can either give it the cold shoulder when we pass it in the hall, or we can leave space for the "skipped" terms but not write them out.
Find using polynomial long division.
First we set up the long division problem. We make sure the terms of the polynomials are written in descending order by exponent, and we leave space in the dividend for the x4 and x3 terms, even though we don't have any. It's like leaving a place at the table for your dad, even though he's currently in Baltimore on business. Dressing up your stuffed rabbit in a fedora and glasses and seating him in your dad's chair might be going too far, though.
We'll work our way through the long division. First, figure out how many times the first term of the divisor goes into the first term of the dividend, perform the necessary subtraction, and bring down the remaining terms of the polynomial:
Now we see how many times the first term of the divisor goes into the first term of our new polynomial, and carry out the appropriate subtraction:
The final answer is .
Check the answer to the above example by multiplying (2x4 + 3) by (4x2 + x). You should get 8x6 + 12x2 + 2x5 + 3x. If you don't, go back to square one. Collect $200.