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Question
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
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Solution
x = − 5 + 4i
∴ x + 5 = 4i
∴ (x + 5)2 = 16i2
∴ x2 + 10x + 25 = − 16 ...[∵ i2 = − 1]
∴ x2 + 10x + 41 = 0 ...(i)
x2 − x + 4
∵ `x^2 + 10x + 41")"overline(x^4 + 9x^3 + 35x^2 - x + 164`
x4 + 10x3 + 41x2
− − −
− x3 − 6x2 − x + 164
− x3 − 10x2 − 41x
+ + +
4x2 + 40x + 164
4x2 + 40x + 164
− − −
0
∴ x4 + 9x3 + 35x2 − x + 164
= (x2 + 10x + 41) (x2 − x + 4)
= 0(x2 − x + 4) ...[From (i)]
∴ x4 + 9x3 + 35x2 − x + 164 = 0
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