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Question
Find the value of 2x4 + 5x3 + 7x2 − x + 41, if x = `-2 - sqrt(3)"i"`
Solution
x = `-2 - sqrt(3)"i"`
∴ x + 2 = `-sqrt(3)"i"`
∴ (x + 2)2 = 3i2
∴ x2 + 4x + 4 = 3(–1) ...[∵ i2 = – 1]
∴ x2 + 4x + 7 = 0 ...(i)
2x2 – 3x + 5
`x^2 + 4x + 7")"overline( 2x^4 + 5x^3 + 7x^2 - x + 41`
2x4 + 8x3 + 14x2
– – –
– 3x3 – 7x2 – x + 41
– 3x3 – 12x2 – 21x
+ + +
5x2 + 20x + 41
5x2 + 20x + 35
– – –
6
Dividend = Divisor x Quotient + Remainder
∴ 2x4 + 5x3 + 7x2 – x + 41
= (x2 + 4x + 7) (2x2 – 3x + 5) + 6
= 0(2x2 – 3x + 5) + 6 ...[From (i)]
= 0 + 6
= 6
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