Saved Bookmarks
| 1. |
If \( x^{2}-x+1 \) divides the polynomial \( x^{n+1}-x^{n}+1 \), then \( n \) must be of the form(a) \( 3 k+1 \)(b) \( 6 k+1 \)(c) \( 6 k-1 \)(d) \( 3 k-1 \) |
|
Answer» For n = 1, xn+1 - xn + 1 = x2 - x + 1 which is exactly divided by x2 - x +1. \(\therefore\) options (c) & (d) are wrong. For n = 4, xn+1 - xn + 1 = x5 - x4 + 1 And \(\frac{x^5-x^4+1}{x^2-x+1}\) = x3 - x - 1 + \(\frac2{x^2-x+1}\) does not divided by x2 - x + 1. For n = 7, xn+1 - xn + 1 = x8 - x7 + 1 And \(\frac{x^8-x^7+1}{x^2-x+1}\) = x6 - x4 - x4 + x + 1 Hence, x2 - x + 1 divides xn+1 - xn + 1 for n = 7 which satisfies by option (b). Hence, for dividing xn+1 - xn + 1 by x2 - x + 1 then n must be of the form 6k + 1. |
|