Using the Remainder Theorem find the remainders obtained when

Algebra (10)

Using the Remainder Theorem find the remainders obtained whenΒ x3 + (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

π‘₯3 + (π‘˜π‘₯ + 8)π‘₯ + π‘˜ = π‘₯3 + π‘˜π‘₯2 + 8π‘₯ + π‘˜

When divided by π‘₯ + 1

Remainder = (-1)3 + π‘˜(-1)2 + 8(-1) + π‘˜

= -1 + π‘˜ - 8 + π‘˜

= 2π‘˜ - 9

When divided by π‘₯ - 2

Remainder = (2)3 + π‘˜(2)2 + 8(2) + π‘˜

= 8 + 4π‘˜ + 16 + π‘˜

= 5π‘˜ + 24

Hence 2π‘˜ - 9 + 5π‘˜ + 24 = 1

7π‘˜ + 15 = 1

7π‘˜ = -14

∴ π‘˜ = -2

Exam Year: 2019