Synthetic DivisionDate: 05/06/2003 at 22:23:17 From: Noel Subject: Algebra (x^2-6x+9)/(x-3) Date: 05/07/2003 at 03:38:15 From: Doctor Luis Subject: Re: Algebra Hi Noel, Here's how you divide a polynomial by (x-a): First draw a grid like this, showing the powers of x and with the coefficients below. x^2 x 1 ------------------- 3 | +1 -6 +9 --|------------------- | If a power of x is missing, make sure you put a 0 as the coefficient. On the side, put the negative of the number you're dividing by. If you're dividing by x-3, put -(-3)=+3. If you're dividing by x+1 put -1 there. Done drawing it? Good. First, bring down the 1. (the first number inside the box) x^2 x 1 ------------------- 3 | +1 -6 +9 --|------------------- | 1 Next, multiply the number you brought down by the number on the side. Since I brought down a 1, I multiply 3*1 = 3. Add this number to the second number (here it's -6), and write that number down (3-6 = -3) in the next spot. x^2 x 1 ------------------- 3 | +1 -6 +9 --|------------------- | 1 -3 Now repeat this process. Multiply the 3 times -3 and add it to the 9. 3*(-3)+9 = 0. x^2 x 1 ------------------- 3 | +1 -6 +9 --|------------------- | 1 -3 0 This last number should be a zero (for our problem it is), if the polynomial x-3 divides x^2-6x+9. If it isn't, then it represents a remainder. This is how you read the answer back. x^2 x 1 ------------------- 3 | +1 -6 +9 --|--------------|---- | 1 -3 | 0 |--------------|---- x 1 You start counting one power of x less on the bottom row. Since we started with x^2, the next lower power is x. You can see the answer is x-3. Remember that the last number is actually a remainder (technically, it represents the -1 power). Therefore, x^2 - 6x + 9 ------------- = x - 3 x - 3 This process is called synthetic division. Let's try a more difficult example that'll teach you more about synthetic division. How about dividing x^3 + 3x + 8 by x+1 ? I won't draw the powers of x this time, because you already know what the numbers in the box mean. You can imagine them to be there in your mind. Note that the 0 is there because there's no x^2 power. | -1 | 1 0 3 8 ---|-----------|---- | 1 -1 4 | 4 See how that worked? Bring down the 1. Multiply by -1 and add to 0, and you get -1. Multiply -1 by -1 and add to 3, you get 4. Multiply 4 by -1 and add to 8, and you get 4. This time, we did get a remainder. Reading the table back, we see that the solution to our problem is x^3 + 3x + 8 4 -------------- = x^2 - x + 4 + -------- x + 1 x + 1 See what happens when you get a remainder term? There's an extra 4/(x+1) term at the end. That's how you write the remainder. The thing at the bottom of the remainder is just the thing you're dividing by. I hope this helped! Let us know if you have any other questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ |
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