# Synthetic Division?

1. Feb 15, 2006

### forevergone

I'm always curious as to how this works? I mean, it's amazing that, even though dividing by binomials, that this sort of mechanism works but I don't understand quite the qwerks behind this.

Can someone fulfill my curiousity?

2. Feb 15, 2006

### 0rthodontist

It's just like regular long division except in regular long division the unknown variable always equals 10 (if you're working in base 10).

What kind of explanation are you looking for, a proof of the algorithm?

3. Feb 15, 2006

### forevergone

Yes. A proof of the algorithm would better help me understand the workings behind this algorithm.

EDIT: Whats the difference between regular long division using base 10 and synthetic division? Don't they both use a decimal based system?

4. Feb 17, 2006

### Cexy

By synthetic division, I take it you mean what I'd call polynomial division?

When you do a long division, you can write

341 into 212784 = 624

as

(3.100 + 4.10 + 1) into (2.100000 + 1.10000 + 2.1000 + 7.100 + 8.10 + 4) equals 6.100 + 2.20 + 4.

Alternatively, you could write it as

(3x2 + 4x + 1) into (2x5 + 1x4 + 2x3 + 7x2 + 8x + 4) = 6x2 + 2x + 4

and then write x=10.

Do you see what Orthodontist means when he says that synthetic division is just like long division, except that the unknown variable is always 10?

5. Feb 18, 2006

### forevergone

Maybe I'm getting the wrong idea from this, but you're saying that if you divide ANY polynomial expression through synthetic division, I would always get 10 as one of my roots?

6. Feb 18, 2006

### HallsofIvy

No, he didn't say anything like that- he said that if you put x= 10 the polynomials you get the numbers in the example.

7. Feb 20, 2006

### ovoleg

synthetic division is for lazy people.

Long division always wins

8. Feb 20, 2006

### forevergone

Not true at all.

Intelligent people always go for the more efficient method of solving problems.

As for putting x = 10 into the polynomials, I don't understand this step:

Can you please tell me how you derive that?

9. Feb 20, 2006

### HallsofIvy

" 3.100+ 4.10+ 1" should be 3*100+ 4*10+ 1 which is, of course, 341.

10. Feb 20, 2006

### Tide

Synthetic division IS long division - only done more efficiently.

11. Feb 20, 2006

### Cexy

Sorry, yeah, I always use a '.' to mean multiplication now. Force of habit etc...