Question

If the polynomial


`{x}^(19) + {x}^(17) + {x}^(13) + {x}^(11) + {x}^(7) + {x}^(5) + {x}^(3)`
is divided by (x² + 1), then the remainder is:
(a) 1
(b) x² + 4
(c) -x
(d) x

Please solve this for me. Spam free answer required. Thank you.​

Answer 1

• (C) - x

Explanation:

Given polynomial:

• x¹⁹ + x¹⁷ + x¹³ + x⁷ + x⁵ + x³

Group as follows:

• (x¹⁹ + x¹⁷) + (x¹³ + x¹¹) + (x⁷ + x⁵) + (x³ + x) - x =
• x¹⁷(x² + 1) + x¹¹(x² + 1) + x¹⁵(x² + 1) + x(x² + 1) - x

As we see all terms have (x² + 1) as factor apart from the last one.

It means the remainder is - x

Correct choice is C