If the polynomial x6 + px5 + qx4 – x2 – x – 3 is divisible by x4

1.	If the polynomial x6 + px5 + qx4 – x2 – x – 3 is divisible by x4 – 1, then the value of p2 + q2 is :(a) 1 (b) 9 (c) 10 (d) 13
Answer» (c) 10 f(x) = x⁶ + px⁵ + qx⁴ – x² – x – 3 = x⁴ . x² + p.x⁴ x + q.x⁴ – x² – x – 3 As (x⁴ – 1) is a factor of f(x), so putting x⁴ = 1, we get x² + px + q – x² – x – 3 = 0 ⇒ (p – 1)x + (q – 3) = 0 ⇒ p – 1 = 0 and q – 3 = 0 ⇒ p = 1 and q = 3. ∴ p² + q² = 1 + 9 = 10.

If the polynomial x6 + px5 + qx4 – x2 – x – 3 is divisible by x4 – 1, then the value of p2 + q2 is :(a) 1 (b) 9 (c) 10 (d) 13

Answer»

(c) 10

f(x) = x⁶ + px⁵ + qx⁴ – x² – x – 3

= x⁴ . x² + p.x⁴ x + q.x⁴ – x² – x – 3

As (x⁴ – 1) is a factor of f(x), so putting x⁴ = 1, we get

x² + px + q – x² – x – 3 = 0

⇒ (p – 1)x + (q – 3) = 0 ⇒ p – 1 = 0 and q – 3 = 0

⇒ p = 1 and q = 3.

∴ p² + q² = 1 + 9 = 10.