Math, asked by bsimran219, 1 month ago

(y" - b") is completely divisible by (y - b), when​

Answers

Answered by avanibaile
0

Answer:

Step 1: putting n=1, we get

x2n−y2n=x2−y2=(x−y)(x+y)

above number is divisible by (x+y). Hence the statement is true n=1

step 2: assuming that for n=m, (x2n−y2n) is divisible by (x+y) then we have

(x2m−y2m)=k(x+y)

Where, k is an integer

step 3: putting n=m+1 we get

x2(m+1)−y2(m+1)

=x2m+2−y2m+2

=x2m+2−x2my2−y2m+2+x2my2

=(x2m+2−x2my2)+(x2my2−y2m+2)

=x2m(x2−y2)+y2(x2m−y2m)

Setting the value of (x2m−y2m) from (1), we get

x2m(x−y)(x+y)+y2k(x+y)

=k(x2m(x−y)+y2)(x+y)

It is clear that the above number is divisible by (x+y). Hence the statement holds for n=m+1

Thus (x2n−y2n) is divisible by (x+y) for all positive integers n≥1

Share Follow

answered

Jul 29 '15 at 12:56

Harish Chandra Rajpoot

35.6k●4343 gold badges●6868 silver badges●103103 bronze badges edited

Jul 29 '15 at 13:29

Up vote

8

Down vote

Hint: Rewrite:

x2k+2−y2k+2=(x2k+2−y2kx2)+(y2kx2−y2k+2)=x2(x2k−y2k)+y2k(x2−y2).

Added: Note it can be proved without induction:

x2n−y2n=(x2)n−(y2)n=(x2−y2)(x2(n−1)+x2(n−2)y2+⋯+x2y2(n−2)+y2(n−1)),

and the first factor is divisible by x+y.

Share Follow

answered

Jul 29 '15 at 12:38

Bernard

165k●99 gold badges●5858 silver badges●151151 bronze badges edited

Jul 29 '15 at 13:23

Up vote

3

Down vote

x2(k+1)−y2(k+1)=x2x2k−y2y2k=x2x2k−x2y2k+x2y2k−y2y2k=x2(x2k−y2k)+(x2−y2)y2k.

The first term is divisible by x+y by the induction hypothesis, and the second as well, by direct factorization.

Share Follow

answered

Jul 29 '15 at 12:47

Yves Daoust

199k●1414 gold badges●120120 silver badges●304304 bronze badges

why must we have that green part? – momokjaaaaa Mar 26 '17 at 10:05

Because it allows the next step. – Yves Daoust Mar 27 '17 at 6:54

Add a comment

Up vote

0

Down vote

For n=k, assume P(k) is true, we have

x2k−y2k=A(x−y)

, where A is a polynomial.

For n=k+1,

x2k+2−yk+2 =x2[A(x−y)+y2k]−y2k+2 =A(x−y)x2+x2y2k−y2k+2 =A(x−y)x2+y2k(x2−y2) =A(x−y)x2+y2k(x−y)(x+y) =(x−y)[Ax2+y2k(x+y)] =B(x−y), where B is a polynomial.

Share Follow

answered

Jul 29 '15 at 12:39

Similar questions