First, set up the long division. Although the dividend skips the terms *x*^{3} and *x*^{2}, what do we do? That's right, we include 'em. Write them as 0*x*^{3} and 0*x*^{2} 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 *x*^{3} column, so it's a good thing we had it. Otherwise, we wouldn't have had anywhere to write -2*x*^{3} 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 3*x*^{2} goes into the first term of our new polynomial, subtract, and bring down terms: By the way, we needed the *x*^{2} column too. Thank goodness we included him. Isn't it nice not having any regrets? Now we see how many times 3*x*^{2} goes into 6*x*^{3}, and so on: Finally, 3*x*^{2} goes into 6*x*^{2} twice: The final answer is . Check the answer to the above example by multiplying (3*x*^{2} – 2) and (3*x*^{3} + 3*x*^{2} + 2*x* + 2). You should get 9*x*^{5} + 9*x*^{4} – 4*x* – 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. |