What is the complete factorization of the polynomial function over the set of complex numbers?

f (x) = x3+3x2+16x+48

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Answer: (x + 3) (x + 4i) (x - 4i)

Explanation:

1) Given: x³ + 3x² + 16x + 48

2) Group terms: (x³ + 3x²) + (16x + 48)

3) Common factors: x² (first group) and 16 (second group):

x² (x + 3) + 16 (x + 3)

4) Common factor x + 3: (x + 3) (x² + 16)

5) Trick: i² = - 1 ⇒ - i² = 1 ⇒ 16 = - 16i² = - (4i) ²

⇒ (x + 3) [ x² - (4i) ² ]

6) Factoring difference of squares: a² - b² = (a + b) (a - b)

⇒ (x + 3) [ x² - (4i) ² ] = (x + 3) (x + 4i) (x - 4i)

Result: (x + 3) (x + 4i) (x - 4i) ← complete factorization