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Question
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Solution
x = 1 + 2i
∴ x − 1 = 2i
∴ (x − 1)2 = 4i2
∴ x2 − 2x + 1 = − 4 ...[∵ i2 = − 1]
∴ x2 − 2x + 5 = 0 ...(i)
x + 4
∵ `x^2 - 2x + 5")"overline(x^3 + 2x^2 - 3x + 21`
x3 − 2x2 + 5x
− + −
4x2 − 8x + 21
4x2 − 8x + 20
− + −
1
∴ x3 + 2x2 − 3x + 21
= (x2 − 2x + 5)(x + 4) + 1
= 0.(x + 4) + 1 ...[From (i)]
= 0 + 1
∴ x3 + 2x2 − 3x + 21 = 1
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