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Question
Find the value of: x3 – 3x2 + 19x – 20, if x = 1 – 4i
Solution
x = 1 – 4i
∴ x – 1 = – 4i
∴ (x – 1)2 = 16i2
∴ x2 – 2x + 1 = – 16 ...[∵ i2 = – 1]
∴ x2 – 2x + 17 = 0 ...(i)
x – 1
`x^2 – 2x + 17")"overline(x^3 - 3x^2 + 19x - 20)"`
x3 – 2x2 + 17x
– + –
– x2 + 2x – 20
– x2 + 2x – 17x
– + –
– 3
∴ x3 – 3x2 + 19x – 20
= (x2 – 2x + 17) (x – 1) – 3
= 0(x – 1) – 3 ...[From (i)]
= 0 – 3
∴ x3 – 3x2 + 19x – 20 = – 3
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