If the polynomial x19 + x17 + x13 + x11 + x7 + x5 + x3 is divided by (

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

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»

(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).\(x\)

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