# How does synthetic division work?

I've read that we can divide any polynomial by a linear polynomial by synthetic division considerably faster than that by long division method.

Now, I've learnt the steps to do so but I don't quite understand how it works.

Because here rather than dividing by the factor we are dividing by the zero ( i.e.* root*) and apparently these two cases are quite different.

But, in a way, the steps involved in this method seem to be equivalent to that in the long division but I haven't been to fully grasp how and why that is so.

Bonus Question:- Also, is there a similar relatively easy method for dividing by higher degrees polynomials too?

• If you know $f(x) = \sum_{n=0}^N a_n x^n$ how do you compute the $b_n$ such that $f(x+c) = \sum_{n=0}^N b_n x^n$ ? – reuns Sep 24 '17 at 17:54
• Have you done any research? – gen-z ready to perish Sep 24 '17 at 18:05
• @Chase, I've learnt how to apply the method. On a website (Purplemath) they compared it to long division method visually, without an explanation. From that I got that these two methods are equivalent in some way but I don't know exactly how. – Mr Reality Sep 24 '17 at 18:08
• Can anyone attempt to answer the question? – Mr Reality Sep 25 '17 at 10:58
• @reuns, I tried but I don't know how to compute $b_n$ there. – Mr Reality Sep 25 '17 at 17:09