Question

If the polynomial x¹⁹ + x¹⁷ + x¹³ + x¹¹ + x⁷ + x⁵ + x³ is divided by (x² + 1), then the remainder is :

(a) 1 

(b) x² + 4 

(c) –x 

(d) x

Answer 1

(c)  – \(x\).

f(x) = x¹⁹ + x¹⁷ + x¹³ + x¹¹ + x⁷ + x⁵ + x³ 

Putting x² = – 1, we get 

f(x) = (x²)⁹.\(x\) + (x²)⁸.\(x\) + (x²)⁶.\(x\) + (x²)⁵.\(x\) + (x²)².\(x\) + x².\(x\) 

= (–1)⁹\(x\) + (–1)⁸.\(x\) + (–1)⁶.^x + (–1)⁵.\(x\) + (–1)².\(x\) + (–1).\(x\) 

= – \(x\) + \(x\) + \(x\) – \(x\) + \(x\) – \(x\) = – \(x\).