Question

Which one of the following is divisible by (1 + a + a⁵) and (1 + a⁴ + a⁵) individually ?

(a) (a² + a + 1) (a³ + a² + 1) (a³ + a + 1) 

(b) (a⁴ – a + 1) (a³ + a² + 1) (a³ + a – 1) 

(c) (a⁴ + a + 1) (a³ – a² + 1) (a³ + a + 1) 

(d) (a² + a + 1 ) (a³ – a² + 1) (a³ – a + 1)

Answer 1

(b) (a⁴ – a + 1) (a³ + a² + 1) (a³ + a – 1) 

The given expression has to be divided by (1 + a + a5) and (1 + a⁴ + a⁵) individually, so highest power of a is 5 + 5 = 10, followed by 5 + 4 = 9, and both are positive.

In options (a) and (d) the highest power of a is 8, hence these options are not acceptable.

The only choices are (b) and (c), but in option (c) a¹⁰ is positive but a⁹ is negative and in (b) both a¹⁰ and a⁹ are positive. Hence (b) is the correct option.