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Find the value of x3 - x2 + x + 46, if x = 2 + 3i

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x = 2 + 3i

x – 2 = 3i

(x – 2)2 = 9i2

x2 – 4x + 4 = 9(-1) …..[∵ i2 = -1]

x2 – 4x + 13 = 0 …..(i)

Dividend = Divisor × Quotient + Remainder

∴ x3 – x2 + x + 46 = (x2 – 4x + 13) (x + 3) + 7

= 0(x + 3) + 7 …..[from(i)]

= 7

Alternate Method:

x = 2 + 3i

α = 2 + 3i, \(\bar \alpha\) = 2 – 3i

 α \(\bar \alpha\)= (2 + 3i)(2 – 3i)

= 4 – 6i + 6i – 9i2

= 4 - 9(-1)

= 4 + 9

= 13

α + \(\bar \alpha\) = 2 + 3i + 2 - 3i = 4

∴ Standard form of quadratic equation,

x2 – (Sum of roots) x + Product of roots = 0

x2 – 4x + 13 = 0

Dividend = Divisor × Quotient + Remainder

∴ x3 – x2 + x + 46 = (x2 – 4x + 13).(x + 3) + 7

= 0(x + 3) + 7 …..[From (i)]

= 7

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