Using the Remainder Theorem find the remainders obtained when xy + (kx

1.	Using the Remainder Theorem find the remainders obtained when xy + (kx + 8)x + k is divided by x + 1 and x – 2. Hence, find k if the sum of the two remainders is 1.
Answer» Given polynomial is p(x) = x³ +(kx + 8)x + k g(x) = x + 1 ∴ R₁ = P(-1) = (-1)³ + {k{-1} + 8} (-1) + k = -1 + k – 8 + k = 2k – 9 h(x) = x – 2 ∴ R₂ = P(2) = (2)³ + (2k + 8)² + k = 8 + 4k + 16 + k 5k + 24 Now, R₁ + R₂ = 1 ⇒ 2k – 9 + 5k + 24 = 1 ⇒ 7k = 1 + 9 – 24 ⇒ 7k = – 14 ⇒ k = – 2

Using the Remainder Theorem find the remainders obtained when xy + (kx + 8)x + k is divided by x + 1 and x – 2. Hence, find k if the sum of the two remainders is 1.

Answer»

Given polynomial is p(x) = x³ +(kx + 8)x + k

g(x) = x + 1

∴ R₁ = P(-1)

= (-1)³ + {k{-1} + 8} (-1) + k

= -1 + k – 8 + k

= 2k – 9

h(x) = x – 2

∴ R₂ = P(2) = (2)³ + (2k + 8)² + k

= 8 + 4k + 16 + k

5k + 24

Now, R₁ + R₂ = 1

⇒ 2k – 9 + 5k + 24 = 1

⇒ 7k = 1 + 9 – 24

⇒ 7k = – 14

⇒ k = – 2