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Question
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
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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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