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.
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.
