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Question
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
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Solution
x = `10/(3 - "i")`
∴ x = `(10(3 + "i"))/((3 - "i")(3 + "i")`
= `(10(3 + "i"))/(9 - "i"^2)`
= `(10(3 + "i"))/(9 - (- 1)` ...[∵ i2 = –1]
= `(10(3 + "i"))/10`
∴ x = 3 + i
∴ x – 3 = i
∴ (x – 3)2 = i2
∴ x2 – 6x + 9 = –1 ...[∵ i2 = –1]
∴ x2 – 6x + 10 = 0 ...(i)
x + 1
`x^2 – 6x + 10")"overline(x^3 - 5x^2 + 4x + 8)"`
x3 – 6x2 + 10x
– + –
x2 – 6x + 8
x2 – 6x + 10
– + –
– 2
∴ x3 – 5x2 + 4x + 8
= (x2 – 6x + 10)(x + 1) – 2
= 0 (x + 1) – 2 ...[From (i)]
= 0 – 2
∴ x3 – 5x2 + 4x + 8 = – 2.
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