Answer :
To determine which polynomial has exactly one real root, we need to look at the options given:
Let's consider the polynomials:
1. Option A: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
2. Option B: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
3. Option C: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
4. Option D: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
For a cubic polynomial to have exactly one real root, it must have either two complex conjugate roots or the root must have multiplicity three (which is rare for these kinds of expressions and usually appears in forms like [tex]\((x-a)^3\)[/tex]).
Analyzing the Options:
- Option A:
Since the coefficients are symmetrical and positive, this polynomial's graph will likely be always increasing. It's common for such polynomials to have one real root and no turning points. Solving or estimating could suggest the nature of the roots, but the graph tends to cross the x-axis once.
- Option B:
The changes in signs indicate it could have multiple real roots. Checking further with derivatives or fact-checking roots will likely reveal more than one real root.
- Option C:
The positive coefficients and similar structure might point to a similar case as Option A, but examining further, this configuration typically yields more than one real root.
- Option D:
The change in sign and the structure suggest it could have one real root. Solving it directly or testing via derivatives would show that it's possible for others to be complex as [tex]\( x^3 + 3x^2 - 9x - 27 = (x - a)(x^2 + bx + c) \)[/tex].
Conclusion:
Upon analysis, Option D is a likely candidate for having exactly one real root due to the structure and behavior of the polynomial in that context, often contrasting where one real and two complex roots appear.
The correct answer is:
D. [tex]\(F(x) = x^3 + 3x^2 - 9x - 27\)[/tex]
Let's consider the polynomials:
1. Option A: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
2. Option B: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
3. Option C: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
4. Option D: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
For a cubic polynomial to have exactly one real root, it must have either two complex conjugate roots or the root must have multiplicity three (which is rare for these kinds of expressions and usually appears in forms like [tex]\((x-a)^3\)[/tex]).
Analyzing the Options:
- Option A:
Since the coefficients are symmetrical and positive, this polynomial's graph will likely be always increasing. It's common for such polynomials to have one real root and no turning points. Solving or estimating could suggest the nature of the roots, but the graph tends to cross the x-axis once.
- Option B:
The changes in signs indicate it could have multiple real roots. Checking further with derivatives or fact-checking roots will likely reveal more than one real root.
- Option C:
The positive coefficients and similar structure might point to a similar case as Option A, but examining further, this configuration typically yields more than one real root.
- Option D:
The change in sign and the structure suggest it could have one real root. Solving it directly or testing via derivatives would show that it's possible for others to be complex as [tex]\( x^3 + 3x^2 - 9x - 27 = (x - a)(x^2 + bx + c) \)[/tex].
Conclusion:
Upon analysis, Option D is a likely candidate for having exactly one real root due to the structure and behavior of the polynomial in that context, often contrasting where one real and two complex roots appear.
The correct answer is:
D. [tex]\(F(x) = x^3 + 3x^2 - 9x - 27\)[/tex]
