How Does Synthetic Division Work?

[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?

 

[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?

 

[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?

 

[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?

 

[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?

 

[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.

 

[ovoleg]

synthetic division is for lazy people.

Long division always wins

 

[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:

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.

Can you please tell me how you derive that?

 

[HallsofIvy]

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

 

[Tide]

synthetic division is for lazy people.

Long division always wins

Synthetic division IS long division - only done more efficiently.

 

[Cexy]

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

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