What is/are the factors of (x29 – x24 + x13 – 1) ?(a) (x – 1) on

1.	What is/are the factors of (x29 – x24 + x13 – 1) ?(a) (x – 1) only (b) (x + 1) only (c) (x – 1) and (x + 1) (d) Neither (x – 1) nor (x + 1)
Answer» (a) (x – 1) only For (x – 1) to be a factor of the given expression, the value of expression at x = 1 is (1)²⁹ – (1)²⁴ + (1)¹³ – 1 = 1 – 1 + 1 – 1 = 0 ∴ (x – 1) is a factor of x²⁹ – x²⁴ + x¹³ – 1 Similarly for (x + 1) to be the factor, the value of expression at x = – 1 is (–1)²⁹ – (–1)²⁴ + (–1)¹³ – 1 = – 1 – 1 – 1 – 1 = – 4 ≠ 0 ∴ (x + 1) is not a factor of x²⁹ – x²⁴ + x¹³ – 1. Hence, (a) is the correct option.

What is/are the factors of (x29 – x24 + x13 – 1) ?(a) (x – 1) only (b) (x + 1) only (c) (x – 1) and (x + 1) (d) Neither (x – 1) nor (x + 1)

Answer»

(a) (x – 1) only

For (x – 1) to be a factor of the given expression, the value of expression at x = 1 is

(1)²⁹ – (1)²⁴ + (1)¹³ – 1 = 1 – 1 + 1 – 1 = 0

∴ (x – 1) is a factor of x²⁹ – x²⁴ + x¹³ – 1

Similarly for (x + 1) to be the factor, the value of expression at x = – 1 is

(–1)²⁹ – (–1)²⁴ + (–1)¹³ – 1 = – 1 – 1 – 1 – 1 = – 4 ≠ 0

∴ (x + 1) is not a factor of x²⁹ – x²⁴ + x¹³ – 1.

Hence, (a) is the correct option.