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(Thread should be named: Question about proof of the division algorithm, sorry about that)

Hi,

I am reading this proof of the division article:

http://xmlearning.maths.ed.ac.uk/lecture_notes/polynomials/division_algorithm/division_algorithm.php" [Broken]

I will write some of it here in case you don't want to click on any links.

Theorem: Let [tex]f,g \in R[X] [/tex] be polynomials with [tex]g \neq [/tex].

Then there exist unique polynomials [tex]q,r \in R[X] [/tex] such that [tex] f=gq+r [/tex] and [tex]deg(r)<deg(g) [/tex].

Proof: We first prove existence of [tex]q,r[/tex].

If [tex]deg(g)>deg(f)[/tex] then we set [tex]q=0[/tex] and [tex]r=f[/tex].

Otherwise, [tex]deg(f) \geq deg(g)[/tex]

Let,

[tex] f=a_0+...+a_mX^m[/tex] where [tex]a_m \neq 0[/tex].

and

[tex] g=b_0+...+b_nX^n[/tex] where [tex]b_n \neq 0[/tex].

Define the integer [tex]d=m-n \geq 0[/tex]. We will use induction in [tex]d[/tex].

I will stop there.

How come they set [tex] q=\frac{a_m}{b_m}[/tex]?

Since [tex]f(x)=q(x)g(x)+r(x)[/tex] isn't then

[tex]q(x)=\frac{f(x)}{g(x)} - \frac{r(x)}{g(x)}[/tex] ?

Thank you for your time.

Hi,

I am reading this proof of the division article:

http://xmlearning.maths.ed.ac.uk/lecture_notes/polynomials/division_algorithm/division_algorithm.php" [Broken]

I will write some of it here in case you don't want to click on any links.

Theorem: Let [tex]f,g \in R[X] [/tex] be polynomials with [tex]g \neq [/tex].

Then there exist unique polynomials [tex]q,r \in R[X] [/tex] such that [tex] f=gq+r [/tex] and [tex]deg(r)<deg(g) [/tex].

Proof: We first prove existence of [tex]q,r[/tex].

If [tex]deg(g)>deg(f)[/tex] then we set [tex]q=0[/tex] and [tex]r=f[/tex].

Otherwise, [tex]deg(f) \geq deg(g)[/tex]

Let,

[tex] f=a_0+...+a_mX^m[/tex] where [tex]a_m \neq 0[/tex].

and

[tex] g=b_0+...+b_nX^n[/tex] where [tex]b_n \neq 0[/tex].

Define the integer [tex]d=m-n \geq 0[/tex]. We will use induction in [tex]d[/tex].

**Base step:**Let [tex]d=0[/tex], then [tex]m=n[/tex]. We set [tex] q=\frac{a_m}{b_m}[/tex].I will stop there.

How come they set [tex] q=\frac{a_m}{b_m}[/tex]?

Since [tex]f(x)=q(x)g(x)+r(x)[/tex] isn't then

[tex]q(x)=\frac{f(x)}{g(x)} - \frac{r(x)}{g(x)}[/tex] ?

Thank you for your time.

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