Is it true that ...? #5

Algebra Level 2

x n − y n = ( x − y ) ( x n − 1 + x n − 2 y + x n − 3 y 2 + ⋯ + y n − 1 ) x^n-y^n = (x-y)\left(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + \cdots + y^{n-1}\right)

Is the above true for all integers n ≥ 2 n \ge 2 ?

Yes No

Samuel Nascimento by Samuel Nascimento , 24, Brazil

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2 solutions

Chew-Seong Cheong
Jul 21, 2019

RHS = ( x − y ) ( x n − 1 + x n − 2 y + x n − 3 y 2 + ⋯ + y n − 1 ) = x ( x n − 1 + x n − 2 y + x n − 3 y 2 + ⋯ + y n − 1 ) − y ( x n − 1 + x n − 2 y + x n − 3 y 2 + ⋯ + y n − 1 ) = ( x n + x n − 1 y + x n − 2 y 2 + ⋯ + x y n − 1 ) − ( x n − 1 y + x n − 2 y 2 + ⋯ + x y n − 1 + y n ) = x n − y n = LHS \begin{aligned} \text{RHS} & = (x-y)\left(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + \cdots + y^{n-1}\right) \\ & = x\left(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + \cdots + y^{n-1}\right) - y\left(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + \cdots + y^{n-1}\right) \\ & = \left(x^n + {\color{#D61F06} x^{n-1}y + x^{n-2}y^2 + \cdots + xy^{n-1}} \right) - \left({\color{#D61F06} x^{n-1}y + x^{n-2}y^2 + \cdots + xy^{n-1}} + y^n \right) \\ & = x^n - y^n = \text{LHS} \end{aligned}

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@Samuel Nascimento , no need to mention x , y ∈ R x, y \in \mathbb R , because the equation is also true for complex values of x x and y y . You can start LaTex code by using \ ( \backslash( ... \ ) \backslash) or \ [ \backslash[ ... \ ] \backslash] . For example: \ ( \backslash( x, y \in \mathbb R \ ) \backslash) x , y ∈ R x, y \in \mathbb R and \ ( \backslash( a^\frac 12, \sqrt[3]{x^2+y^2}, \frac \pi 2, \dfrac {\sqrt 3}2, \tan x, \sin x, \cos x, \ln x, \int 0^\frac \pi 2, \sum {k=1}^\infty, \displaystyle \frac \pi 2, \int 0^\frac \pi 2, \sum {k=1}^\infty, \ ) \backslash) a 1 2 , x 2 + y 2 3 , π 2 , 3 2 , tan ⁡ x , sin ⁡ x , cos ⁡ x , ln ⁡ x , ∫ 0 π 2 , ∑ k = 1 ∞ , π 2 , ∫ 0 π 2 , ∑ k = 1 ∞ a^\frac 12, \sqrt[3]{x^2+y^2}, \frac \pi 2, \dfrac {\sqrt 3}2, \tan x, \sin x, \cos x, \ln x, \int_0^\frac \pi 2, \sum_{k=1}^\infty, \displaystyle \frac \pi 2, \int_0^\frac \pi 2, \sum_{k=1}^\infty .

Chew-Seong Cheong - 1 year, 10 months ago
Hana Wehbi
Jul 21, 2019

The title need to be edited. There is two is it true #3

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