Answer :
To determine which polynomial is a 3rd degree polynomial with exactly 1 real root, we need to analyze the root behavior of each polynomial. A 3rd degree polynomial can have up to 3 real roots, which could be distinct or repeated. Let's go through each option:
1. Option A: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- This polynomial has 3 real roots, all of which are equal to -3.
2. Option B: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- This polynomial has exactly 1 real root.
3. Option C: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- This polynomial has 3 real roots, two of which are -3 and one is 3.
4. Option D: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial has 3 real roots, all equal to 3.
From the analysis above, we can see that:
- Option B ([tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]) has exactly 1 real root.
Therefore, the correct option is B.
1. Option A: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- This polynomial has 3 real roots, all of which are equal to -3.
2. Option B: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- This polynomial has exactly 1 real root.
3. Option C: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- This polynomial has 3 real roots, two of which are -3 and one is 3.
4. Option D: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial has 3 real roots, all equal to 3.
From the analysis above, we can see that:
- Option B ([tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]) has exactly 1 real root.
Therefore, the correct option is B.
