Answer :
To determine which of the given polynomials is a 3rd degree polynomial with exactly 1 real root, let's examine each option:
1. Option A: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- This polynomial is a 3rd degree polynomial.
- It has 2 real roots.
2. Option B: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial is also a 3rd degree polynomial.
- It has exactly 1 real root.
3. Option C: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- This polynomial is a 3rd degree polynomial.
- It has 3 real roots.
4. Option D: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- This polynomial is a 3rd degree polynomial.
- It has exactly 1 real root.
From this analysis, Option B and Option D are both 3rd degree polynomials with exactly 1 real root. However, based on the provided context, we are looking for where this happens strictly within the mathematical framework of this assessment. Thus, either polynomial could be the answer, but traditionally, scenarios with solutions like these usually indicate that additional graph behavior like a change of direction or diving and emerging behavior (considering one might inherently go complex for instance) could lead you to B as a conventional choice given problems of nature of such parameters, if there was typically a follow-up context clarifying trials further.
In general classroom questions though, either option B or D is correct unless noted otherwise for specific criteria beyond roots alone. Such scenarios are elaborated based on class specificity or textbooks indicating potential overlapping interpretations outside standard assumptions.
1. Option A: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- This polynomial is a 3rd degree polynomial.
- It has 2 real roots.
2. Option B: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial is also a 3rd degree polynomial.
- It has exactly 1 real root.
3. Option C: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- This polynomial is a 3rd degree polynomial.
- It has 3 real roots.
4. Option D: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- This polynomial is a 3rd degree polynomial.
- It has exactly 1 real root.
From this analysis, Option B and Option D are both 3rd degree polynomials with exactly 1 real root. However, based on the provided context, we are looking for where this happens strictly within the mathematical framework of this assessment. Thus, either polynomial could be the answer, but traditionally, scenarios with solutions like these usually indicate that additional graph behavior like a change of direction or diving and emerging behavior (considering one might inherently go complex for instance) could lead you to B as a conventional choice given problems of nature of such parameters, if there was typically a follow-up context clarifying trials further.
In general classroom questions though, either option B or D is correct unless noted otherwise for specific criteria beyond roots alone. Such scenarios are elaborated based on class specificity or textbooks indicating potential overlapping interpretations outside standard assumptions.
