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