Basic properties of numbers proof

In summary, the conversation discusses different methods for proving the equation x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}). These include distributing the right part, using polynomial long division, and using the formula for the sum of a geometric series. The conversation also mentions the possibility of using induction to prove the equation.
  • #1
Magicalz
3
0

Homework Statement


Prove the following:

[tex] x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}) [/tex]

The Attempt at a Solution


Ugh I just tried to distribute the right part:

[tex]
\begin{equation*}
\begin{split}
\x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}) \\
&= \(x - y)x^{n-1} + (x - y)x^{n-2}y + ... + (x - y)xy^{n-2} + (x - y)y^{n-1} \\
&= x^n - x^{n-1}y + x^{n-1}y - x^{n-2}y^2 + ... + x^2y^{n-2} - xy^{n-1} + xy^n-1 - y^n
\end{split}
\end{equation*}
[/tex]

well the terms that are visible (that are written down) really do cancel out (except of course x^n and y^n) however I guess what I'm having trouble grasping with is what constitutes the "proof" part of this...I mean yes it seems likely that the terms inbetween x^n and y^n are cancelling each other out, but can we be absolutely sure? the dots ... mean continue the pattern,but I guess what I'm wondering is,is what I'm doing by distributing the right side constituting the "proof"? Any help is appreciated thankyou.
 
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  • #2
Do you know anything about polynomial long division ? If so, regard the x^n -y^n and x-y as polynomials in the "x" variable. EDIT: [/tex]
 
  • #3
Hi dextercioby! Thanks for the quick reply. Hmm I don't remember polynomial long division can you show the steps for this particular question? thanks!
 
  • #4
Yes, you can prove it the way you did. Show that the RHS equals the LHS by performing the multiplications in the RHS. Totally okay. Or you can do it the way i suggested.

[tex] \frac{x^{n}-y^{n}}{x-y}=x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1} [/tex]

By polynomial long division you can show that the LHS equals the RHS.
 
  • #5
Hey dextercioby thanks for the help, can you give me a short reminder of how to do polynomial long division??

*oh and just another question, how come on my above attempted solution part, the line is so long, but i put the coding \\ in the latex but there seems to be no new line thing? can anyone help me thankyou
 
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  • #7
You could also try induction, though its quite unnecessary.
 
  • #8
[tex] \frac{x^{n}-y^{n}}{x-y}=x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1} [/tex]

Treat the RHS as a geometric series, first term [tex]x^{n-1}[/tex], common ratio [tex]\frac{y}{x}[/tex]. Sum. Simplify.
 
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  • #9
Kind of circular unless you know of a proof for the formula of the sum of a geometric series that doesn't exploit this identity, which i don't >.<
 
  • #10
Gib Z said:
Kind of circular unless you know of a proof for the formula of the sum of a geometric series that doesn't exploit this identity, which i don't >.<

I suppose you're right - but the question never stated what you could or could not assume. In any case, in addition to your suggestion of induction, it's just another approach one could use.

The proof for GP sum that I know (in fact I found it myself as a kid) is the obvious Sn = ..., rSn = ..., take the difference, etc.

One could also adapt that method to this, although it takes a little more intuition to see that one needs to multiply by (y/x) to get the result.
 

1. What are the basic properties of numbers?

The basic properties of numbers include commutative property, associative property, distributive property, identity property, and inverse property. These properties help us understand how numbers work and how we can manipulate them in equations and expressions.

2. What is the commutative property of addition and multiplication?

The commutative property states that the order of numbers in addition or multiplication does not affect the result. For example, a + b = b + a and a * b = b * a. This property is helpful when adding or multiplying numbers in any order.

3. How does the associative property work?

The associative property states that the grouping of numbers in addition or multiplication does not affect the result. For example, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c). This property is useful when working with long expressions and allows us to change the grouping of numbers without changing the result.

4. What is the identity property of addition and multiplication?

The identity property states that the sum of any number and 0 is equal to the original number, and the product of any number and 1 is equal to the original number. For example, a + 0 = a and a * 1 = a. This property is important because it helps us solve equations and simplify expressions.

5. How does the inverse property work?

The inverse property states that the sum of a number and its additive inverse (or negative) is equal to 0, and the product of a number and its multiplicative inverse (or reciprocal) is equal to 1. For example, a + (-a) = 0 and a * (1/a) = 1. This property is useful when solving equations and working with fractions.

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