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Question
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
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Solution
x = 2 + 3i
∴ x – 2 = 3i
∴ `(x - 2)^2 = (3i)^2`
∴ (x – 2)2 = 9i2
∴ x2 – 4x + 4 = 9(– 1) ...[∵ i2 = – 1]
∴ x2 – 4x + 13 = 0 ...(i)
x + 3
`x^2 – 4x + 13")"overline(x^3 - x^2 + x + 46)"`
x3 – 4x2 + 13x
– + –
3x2 – 12x + 46
3x2 – 12x + 39
– + –
7
∴x3 – x2 + x + 46
= (x2 – 4x + 13)(x + 3) + 7
= 0(x + 3) + 7 ...[From (i)]
= 7.
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