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Question
Using the remainder theorem, find the remainders obtained when 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.
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Solution
Remainder theorem :
Dividend = Divisors × Quotient + Remainder
∴ Let f(x) = x3 + (kx + 8 )x + k
= x3 + kx2 + 8x + k
Dividing f(x) by x + 1 gives remainder as R1
∴ f (−1) = R1
Also, f(2) = R2
∴ f (−1) = (−1)3 + k(−1)2 + 8(−1) + k
= −1 + k − 8 + k
= 2k − 9 = R1
f(2) = (2)3 + k (2)2 + 8(2) + k
= 8 + 4k + 16 + k
= 5k + 24 = R2
Also,Sum of remainders = R1 + R2 = 1
∴ (2k − 9) + (5k +24) = 1
7k + 15 = 1
7k = −14
k = −2
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