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Find factors of $z^{3} - 10 z^{2} + 58 z - 136 = 0$ .

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(z - 4), (z - 3 - 5 i), (z - 3 + 5 i) = 0

(z - 2), (z - 3 - 5 i) =, z - 4 + 6 i = 0

Both 1) and 2)

None of these

answer is A.

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Detailed Solution

z = 1

f (z) = z^{3} - 10 z^{2} + 58 z - 136

f (z) = {(1)}^{3} - 10 {(1)}^{2} + 58 (1) - 136

f (z) = 1 - 10 + 58 - 136

f (z) = - 87

f (z) \neq 0

So,

z - 1 = 0

is not a factor.
When

z = 4

f (z) = z^{3} - 10 z^{2} + 58 z - 136

f (z) = 0

So, equation can be written as

z^{3} - 4 z^{2} - 6 z^{2} + 24 z + 34 z - 136 = 0

z^{2} (z - 4) - 6 z (z - 4) + 34 (z - 4) = 0

f (z) = (z - 4) (z^{2} - 6 z + 34)

By complete the square method,

z = \frac{- (- 6) \pm \sqrt{{(- 6)}^{2} - 4 \times 1 \times 34}}{2 (1)}

z = \frac{6 \pm \sqrt{- 100}}{2}

z = 3 \pm 5 i

So,

(z - 4) (z - 3 - 5 i) (z - 3 + 5 i) = 0

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