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## Main Question or Discussion Point

So I was trying to understand Horner's method. I understand that you can take a polynomial and factor out the x's and re write it as multiple linear functions recursively plugged into each other and that this makes evaluating a polynomial easier because you just evaluate a linear function multiple times. But

I was reading the

-What does this have to do with synthetic division? I know what synthetic division is, I just don't see the conection.

-What does this that to do with Newtons method for calculating the root of a function? I know that newtons method also involves recursive evaluation where the previous result is the new input. Newtons method comes from the linearization, but besides that similarity, I also do not see the connection here.

-After the words

-And why replace the coefficient a with b ?

Also I was watching

And finally, I don't get why I don't get this, its all just elementary operations!

I was reading the

__Wikipedia page__and I do not understand:-What does this have to do with synthetic division? I know what synthetic division is, I just don't see the conection.

-What does this that to do with Newtons method for calculating the root of a function? I know that newtons method also involves recursive evaluation where the previous result is the new input. Newtons method comes from the linearization, but besides that similarity, I also do not see the connection here.

-After the words

*under the**Now, it can be proven that;***section, I do not understand where it gets that equation from? If b_0 is p(x_0) then if you rearrange the terms you have slope equals that polynomial of b's , which I don't understand why they would set that equal to the slope?****Description of the algorithm**-And why replace the coefficient a with b ?

Also I was watching

__this video__on Horner's method. At 2:47 , the last bullet point, the representation as a piece wise defined function I do not understand.And finally, I don't get why I don't get this, its all just elementary operations!