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Answer :
- Option A has one real root.
- Options B and C have one real root with multiplicity 3.
- Option D has two distinct real roots.
- Therefore, the answer is A. $\boxed{A}$
### Explanation
1. Problem Analysis
We are given four 3rd degree polynomials and we want to find the one with exactly one real root.
2. Analyzing Option A
A. $F(x)=x^3+3 x^2+9 x+27$. The approximate real root is $x=-3$.
3. Analyzing Option B
B. $F(x)=x^3+9 x^2+27 x+27$. The approximate real roots are $x=-3.02, x=-3, x=-2.98$.
4. Analyzing Option C
C. $F(x)=x^3-9 x^2+27 x-27$. The approximate real roots are $x=2.98, x=3, x=3.02$.
5. Analyzing Option D
D. $F(x)=x^3+3 x^2-9 x-27$. The approximate real roots are $x=-3, x=3$.
6. Conclusion
From the results of the calculations, we can see that option A, $F(x)=x^3+3 x^2+9 x+27$, has only one real root, $x=-3$. Options B and C have three very close real roots, which suggests they have one real root with multiplicity 3. Option D has two distinct real roots. Therefore, the correct answer is A.
### Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves and surfaces in computer graphics, to describe the motion of objects in physics, and to model growth and decay in biology.
- Options B and C have one real root with multiplicity 3.
- Option D has two distinct real roots.
- Therefore, the answer is A. $\boxed{A}$
### Explanation
1. Problem Analysis
We are given four 3rd degree polynomials and we want to find the one with exactly one real root.
2. Analyzing Option A
A. $F(x)=x^3+3 x^2+9 x+27$. The approximate real root is $x=-3$.
3. Analyzing Option B
B. $F(x)=x^3+9 x^2+27 x+27$. The approximate real roots are $x=-3.02, x=-3, x=-2.98$.
4. Analyzing Option C
C. $F(x)=x^3-9 x^2+27 x-27$. The approximate real roots are $x=2.98, x=3, x=3.02$.
5. Analyzing Option D
D. $F(x)=x^3+3 x^2-9 x-27$. The approximate real roots are $x=-3, x=3$.
6. Conclusion
From the results of the calculations, we can see that option A, $F(x)=x^3+3 x^2+9 x+27$, has only one real root, $x=-3$. Options B and C have three very close real roots, which suggests they have one real root with multiplicity 3. Option D has two distinct real roots. Therefore, the correct answer is A.
### Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves and surfaces in computer graphics, to describe the motion of objects in physics, and to model growth and decay in biology.
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