Polynomial Long DivisionDate: 12/03/2001 at 11:54:18 From: Rachel Subject: Dividing a Polynomial by a Polynomial using long division b^9+6b^6+b^4+9b^3+4b+8 by b^3+4 Why in some questions do you need to add place holders? It has something to do with ascending and descending powers. I am homeschooled and someone said that this was a great site. Please help! Date: 12/03/2001 at 20:14:23 From: Doctor Rick Subject: Re: Dividing a Polynomial by a Polynomial using long division Hi, Rachel. In ordinary long division, you want to be sure to keep the columns lined up correctly - thousands, hundreds, tens, ones - and place holders (0 digits) help you do this. With numbers, such as 203, you need the zero to make it clear that the 2 is in the hundreds place; if you didn't write zeros, "23" could mean 23 or 203 or 230. In polynomial long division, place holders aren't quite so critical, because the power of the variable is written explicitly. It's like writing the number 203 as 2*10^2 + 3; this is perfectly clear without writing 2*10^2 + 0*10 + 3. What you MUST do when doing polynomial long division is to order the terms from highest power to lowest power, just as the digits in a number are ordered from highest power of 10 to lowest. Place holders, while not absolutely necessary, are very useful because they save room for a power of the variable that might appear as you work, where it was previously missing. Let's look at your example. ---------------------------------- b^3 + 4 ) b^9 + 6b^6 + b^4 + 9b^3 + 4b + 8 If we include all the place holders, we get: 1b^6+0b^5+0b^4+2b^3+0b^2+1b+1 ---------------------------------------------- 1b^3+0b^2+0b+4 ) 1b^9+0b^8+0b^7+6b^6+0b^5+1b^4+9b^3+0b^2+4b+8 1b^9+0b^8+0b^7+4b^6 ------------------- 2b^6+0b^5 0b^6+0b^5 --------- 2b^6+0b^5+1b^4 0b^6+0b^5+1b^4 -------------- 2b^6+0b^5+1b^4+9b^3 2b^6+0b^5+0b^4+8b^3 ------------------- 1b^4+1b^3+0b^2 0b^4+0b^3+0b^2 -------------- 1b^4+1b^3+0b^2+4b 1b^4+0b^3+0b^2+4b ---------------- 1b^3+0b^2+0b+8 1b^3+0b^2+0b+4 ------------- 4 The only place holders that you really need are those in the dividend, because you need to reserve space for all the columns. You don't really need place holders in the divisor. In the partial products, you don't need to write the place holders, just remember to put each term in the correct column for its power. You can also learn to skip terms in the quotient. For instance, look at the partial remainder of 2b^6. The next term in the quotient must be a b^3 term in order for the highest term of the partial product to be b^6. Therefore you can bring down the three terms 0b^5+1b^4+9b^3 all at once, and skip the b^5 and b^4 terms in the quotient. You probably do something similar in regular long division. Once you understand how polynomial division works, you can write it like this: b^6 +2b^3 + b+1 -------------------------------------------- b^3+4 ) b^9+0b^8+0b^7+6b^6+0b^5+b^4+9b^3+0b^2+4b+8 b^9 +4b^6 ------------------ 2b^6 +b^4+9b^3 2b^6 +8b^3 ------------------ b^4+ b^3 +4b b^4 +4b ---------------- b^3 +8 b^3 +4 ------------- 4 Do you see how it works now, and why place holders are helpful? Use the place holders at least until you really understand what you're doing. If you aren't sure you do it correctly, please send me a worked-out example like mine (preferably one that you know is wrong). Seeing your work can help me figure out how to help you. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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