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Which of the following options is a 3rd degree polynomial with exactly 1 real root?

A. [tex]F(x) = x^3 + 3x^2 - 9x - 27[/tex]
B. [tex]F(x) = x^3 + 9x^2 + 27x + 27[/tex]
C. [tex]F(x) = x^3 + 3x^2 + 9x + 27[/tex]
D. [tex]F(x) = x^3 - 9x^2 + 27x - 27[/tex]

Answer :

To determine which of the given options is a 3rd-degree polynomial with exactly 1 real root, we analyze each polynomial individually:

1. Option A: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- This polynomial has 2 real roots and 1 complex root.

2. Option B: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- This polynomial has exactly 1 real root.

3. Option C: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- This polynomial also has exactly 1 real root.

4. Option D: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial has exactly 1 real root.

Therefore, the correct options where the polynomial has exactly 1 real root are B, C, and D.

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