Questions about proof of the division article.

In summary, the proof of the division algorithm shows that for any two polynomials f and g, there exist unique polynomials q and r such that f= qg + r and the degree of r is smaller than the degree of g. The proof uses induction and in the base case where the degree of f is equal to the degree of g, the quotient is just a number. In the induction step, the remaining polynomial f_1 is obtained by subtracting the first step of long division from f.
  • #1
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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].

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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  • #2
Dafe said:
(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].


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.
In this "base case", d= 0, the two polynomials have the same degree and so their quotient is a number, not a function of x. For example, if [itex]f(x)= ax^2+ bx+ c[/itex] and [itex]g(x)= ex^2+ fx+ g[/itex] then
[tex]\frac{ax^2+ bx+ c}{ex^2+ fx+ g}[/tex]
is just the number a/e with a first degree remainder.
 
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  • #3
Thank you :blushing:

One more question (kind of the same):
The article goes on with the induction step:

Now we assume that this is true whenever [tex]d<k[/tex] and let [tex]d=k[/tex], so that
[tex]m=n+k[/tex]. Let [tex]f_1=f-(\frac{a_m}{b_n}x^{m-n}g)[/tex].

I do not understand this last step.
Since [tex]\frac{a_m}{b_n}x^{m-n}[/tex] would be the first step of long division, does [tex]f_1[/tex] somehow refer to this?

Feeling quite stupid here. Good thing it's Christmas soon, will drown my sorrows in food and drink.

Thanks again!
 

1. What is the purpose of the division article?

The purpose of the division article is to explain the concept of division and its application in mathematics. It also provides methods for solving division problems and explains the properties of division.

2. How is division different from other mathematical operations?

Division is the process of dividing a number into equal parts or groups. It is the inverse operation of multiplication and is used to find the quotient or the number of times one quantity is contained in another.

3. What are some common strategies for solving division problems?

Some common strategies for solving division problems include using the traditional long division method, using repeated subtraction, and using the multiplication table or inverse operation to find the missing number.

4. How do I know if my division answer is correct?

To check if your division answer is correct, you can use the inverse operation of multiplication to multiply the quotient by the divisor. If the product is equal to the dividend, then your answer is correct.

5. Can division by zero ever be valid?

No, division by zero is considered undefined and therefore not valid in mathematics. This is because any number multiplied by zero will result in zero and it is impossible to determine the quotient when dividing by zero.

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