We're starting to get impatient here. We want to start climbing Polynomial Mountain, but we won't be able to make it to the top without synthetic division, a fancier type of polynomial division. Maybe it's synthetic because it's made out of some space-age polymer?
Sample Problem
Divide x2 + 8x + 15 by x + 3.
To start off, we strip away all of our variables from our polynomial and put the numbers into an open box, like so:
Hey, wait a second. That -3 isn't from the polynomial, and it isn't in the box. What gives?
In synthetic division, we're always dividing a polynomial by x – a. That's where the -3 came from; it's a. We're dividing by x + 3, which is the same thing as x – (-3). Now we can make the division happen.
Step one is to drop the first coefficient (from 1x2) below the line. Next we'll multiply that 1 by a, -3, and put it into the second column.
We get another -3. We'd better hurry before they start multiplying out of control. We'll add it to the 8 already in that column, and put the result down below.
Now we just keep going like we did before: multiply the number at the bottom by our -3, and then add the result to the next column. We'll do this until we run out of numbers.
All gone already? Oh well. Anyway, the rightmost number below the line is the remainder from our division. It's 0, so we have no remainder. That means that x + 3 is a factor of x2 + 8x + 15, since it divides evenly.
Our leftover numbers below the box give us the answer to the division problem. From the right, they start with the remainder, then the constant, in this case 5, and increase in degree as we move left. That means that we have x + 5 as our quotient.
Our final answer is that x2 + 8x + 15 = (x + 3)(x + 5). No, wait, our final answer is (D) a jar of mayonnaise. Did we win, Regis? Did we? Are we going to be a millionaire?
Oh. No. We were right before.
One quick note before we finish up. If our polynomial skips over any powers of x, we need to use a 0 as a placeholder in our synthetic division. For example if we were dividing x3 + 2x + 50 by x + 4, the top line of our synthetic division box would be 1, 0, 2, 50. That's because our polynomial skipped the x2 term, so we're actually dividing x3 + 0x2 + 2x + 50. Got it?
That was the last thing we needed to do before starting up Polynomial Mountain. We hope it goes smoother than a climb of Mount Everest.