Long Division Examples

Use polynomial long division to find .

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.