How Does Synthetic Division Work?

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Discussion Overview

The discussion revolves around the workings of synthetic division, comparing it to traditional long division, particularly in the context of polynomial division. Participants express curiosity about the algorithm, seek proofs, and clarify differences between the two methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses curiosity about how synthetic division works and requests clarification on its mechanisms.
  • Another participant compares synthetic division to regular long division, suggesting that the unknown variable is always set to 10 in synthetic division.
  • A request for a proof of the synthetic division algorithm is made, indicating a desire for deeper understanding.
  • There is a question about the differences between regular long division and synthetic division, particularly regarding their use of a decimal-based system.
  • One participant interprets synthetic division as a form of polynomial division, providing examples of both numerical and polynomial long division.
  • Some participants challenge the idea that synthetic division always yields 10 as a root, clarifying that it is only a specific case when x is set to 10.
  • There is a contention regarding the efficiency of synthetic division versus long division, with differing opinions on which method is superior.
  • Clarifications are made about the notation used in examples, particularly regarding multiplication and the representation of numbers.

Areas of Agreement / Disagreement

Participants express differing views on the efficiency and preference for synthetic division versus long division. There is no consensus on which method is superior, and some misunderstandings about the notation and concepts are present.

Contextual Notes

Some participants exhibit confusion regarding the application of synthetic division and its relationship to long division, particularly in terms of notation and the implications of setting x to 10.

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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?
 
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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?
 
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?
 
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?
 
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?
 
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.
 
synthetic division is for lazy people.

Long division always wins
 
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?
 
" 3.100+ 4.10+ 1" should be 3*100+ 4*10+ 1 which is, of course, 341.
 
  • #10
ovoleg said:
synthetic division is for lazy people.

Long division always wins

Synthetic division IS long division - only done more efficiently.
 
  • #11
HallsofIvy said:
" 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...
 

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