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प्रश्न
Find the value of x3 − x2 + x + 46, if x = 2 + 3i
उत्तर
x = 2 + 3i
∴ x − 2 = 3i
∴ (x − 2)2 = (3i)2
∴ x2 − 4x + 4 = 9i2 = − 9 ...[∵ i2 = − 1]
∴ x2 − 4x + 13 = 0 ...(1)
Now, divide x3 − x2 + x + 46 by x2 − 4x + 13, we get,
x + 3
`x^2 - 4x + 13")"overline( x^3 - x^2 + x + 46`
x3 − 4x2 + 13x
− + −
3x2 − 12x + 46
3x2 − 12x + 39
− + −
7
By division algorithm, we have,
dividend = divisor x quotient + remainder
∴ x3 − x2 + x + 46 = (x2 − 4x + 13)(x + 3) + 7
= 0 × (x + 3) + 7 ...[By (1)]
= 0 + 7
= 7
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