Synthetic/Polynomial division help.

Polynomial...Synthetic division. I just don't get it, which is odd for me, since I usually catch on quickly to most things. Soooo, help? Please?

...if i remember correctly, synthetic division only applies to this situation:

(ax^3+bx^2+cx+d) / (x-e) (or any high root, but the divisor has to be (x-e))

like so:

2x^4-x^3-7x^2-3x+10 / x-2

you should know about long division with polynomials to begin with. You do it like normal division:

________2x^3__________
x-2)2x^4-x^3-7x^2-3x+10
-2x^4+4x^3
---------------------------------
3x^3-7x^2


what you are trying to do is find a term that will get rid of the highest term in the dividend. This being 2x^4. What multiplied by x-2 will give me 2x^4? 2x^3 of course. Then follow down, subtracting. However, since a negative multiplied by a negative makes a positive causes that 4x^3 to become positive. You can see where this is going.

Synthetic is the shortcut.

You take the constant in the divisor and place is next to the the coefficients in the dividend like so:
_____________
2 |2 -1 -7 -3 10
____4__6_-2_-10
2 3 -1 -5 0

Then you take the first coefficient and bring it down. Then you multiply is by the divisor and place it under the next coefficient (2 * 2 = 4), then subtract. Repeat to the end.

Each answer you get is the coeffienct of the answer, one power lower.

Like so: you starting one was 2x^4-x^3-7x^2-3x+10. The answer is 2x^3+3x^2-x-5, with a remainder of 0.

Hope that helps.

I'll have to read over that a few times, but it's coming together. Thx.

Other explanations are still welcome. Not that Lozzie's was bad, but something seen from other points of view are more easily understood (Wow, that was a deep statement!).

Unfortunately, which limited resources (lol internet) it's really hard to explain. If you were sitting in front of me and was able to show you, it would have been easier.

Originally Posted by Lozzie

Unfortunately, which limited resources (lol internet) it's really hard to explain. If you were sitting in front of me and was able to show you, it would have been easier.

So true.

Actually Lozzie spelled it out quite nicely.

But I will give an example as well.

Say you divide 4x^3+2x^2-43x+22 by x-2, you would set x-2 = 0 and get x = 2 then set up the division like Lozzie did.

2| 4 2 -43 22

| 0 8 32 -22
_______________
| 4 16 -11 0

Then you get 4x^2+16x-11

I get it now, this can be locked. Thanks Lozzie and Brolik.