Question

The quotient when 1+x^2+x^4+x^6+...+x^34 is divided by 1+x+x^2+x^3+...+x^17

Answer 1

Quotient is x¹⁷ - x¹⁶ + x¹⁵  - x¹⁴ + x¹³ - x¹² + x¹¹ - x¹⁰ + x⁹ - x⁸ + x⁷ - x⁶ + x⁵ - x⁴ + x³ -x² + x - 1

when 1 + x²  + x⁴  + x⁶  +.........+ x³⁴ divided by 1 + x  + x²  + x³................+ x¹⁷

Step-by-step explanation:

1 + x²  + x⁴  + x⁶  +.............................................+ x³⁴

This is an GP where  

a = 1

r = x²

n = 18

Sum = a(rⁿ - 1)/(r - 1)    = 1 ( x³⁶ - 1)/(x² - 1)

1 + x  + x²  + x³  +.............................................+ x¹⁷

This is an GP where  

a = 1

r = x

n = 18

Sum =   ( x¹⁸ - 1)/(x - 1)

{ ( x³⁶ - 1)/(x² - 1) } / {( x¹⁸ - 1)/(x - 1)}

= (x¹⁸ + 1) / ( x + 1)

Quotient =

x¹⁷ - x¹⁶ + x¹⁵  - x¹⁴ + x¹³ - x¹² + x¹¹ - x¹⁰ + x⁹ - x⁸ + x⁷ - x⁶ + x⁵ - x⁴ + x³ -x² + x - 1

Remainder = 2   by putting x = - 1

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