Can someone explain why (not how!) synthetic division works?

Question

Our teacher asked us this, and I'm not even sure what he means.

Answer 1

In the terminology I was taught, "synthetic" division is a shortcut way to divide a polynomial by a binomial, where the binomial has to be of the form x+a (or x-a).

(To make sure you know what I'm saying, here is the "normal" way to do the division (as opposed to the "synthetic" way). The "normal" division of a polynomial by a binomial is just like long division of one actual number by another. In effect, each term of the binomial (the divisor), and each term of the polynomial (the dividend) is treated as if it were a digit in an actual number. You fill in the answer (the quotient) one term at a time, just as you fill in the answer of a division problem one digit at a time. And you multiply each successive term (digit) of the answer by each term (digit) of the divisor to get a binomial (number) that you subtract from the dividend.)

In synthetic division, instead of writing x+a for the divisor, you write -a. And after you multiply by each term of the answer, you add, instead of subtracting. Also, you don't write in any of the x's, just the coefficients.

So why does it work? All you need to do to answer this is work the same problem two ways: the regular way, and the synthetic way. You should see that each step of the regular division matches a corresponding step of the synthetic division. You just don't have to write as many characters on the page when you do synthetic division.

You'll have to do it in order to understand it.

Answer 2

Try this website: www.lhup.edu/cmorgan/math100syntheticdivision.pdf

There's a section on why it works. I like doing synthetic division, but then I'm strange. LOL

Answer 3

It works as it is short cut for long division . for the steps you can look into

Answer 4

Here's why:

http://www.purplemath.com/modules/synthdiv.htm

Answer 5

Hi, try this da.

http://mathworld.wolfram.com/SyntheticDivision.html

Peace out.