Question

If k and 2k are zeros of f(x) = x3 + 4x2 + 9kx - 90, find k and all three zeros of f(x).
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Answer 1

Step-by-step explanation:

f(x) = x³ + 4x² + 9kx – 90.

f(k) = k³ + 4k² + 9kk – 90 = 0

k³ + 4k² + 9k² – 90 = 0

k³ + 13k² – 90 = 0

k³ + 13k² = 90 Eq. (1)

f(2k) = (2k)³ + 4(2k)² + 9k(2k) – 90 = 0

8k³ + 16k² + 18k² – 90 = 0

8k³ + 34k² – 90 = 0

4k³ + 17k² = 45 Eq. (2)

k³ + 13k² = 90 Eq. (1)

4k³ + 17k² = 45 Eq. (2)

52k³ + 221k² = 585 Eq. (2) × 13

17k³ + 221k² = 1530 Eq. (1) × 17

-----subtract-------------------------

35k³ = –945

k³ = –945/35 = –27

k = –3

2k = –6

Other root = –(–90)/((–3)(–6)) = 5

x = {–3, –6, 5}