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Answer :
To determine which of the given polynomials is a 3rd degree polynomial with exactly one real root, we need to consider each option and find the roots of each polynomial.
Let's go through the polynomials one by one:
Option A: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
This polynomial has been found to have two real roots: [tex]\(-3\)[/tex] and [tex]\(3\)[/tex]. Therefore, it does not have exactly one real root.
Option B: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
This polynomial has exactly one real root: [tex]\(3\)[/tex]. Hence, this is a potential answer as it meets the condition of having exactly one real root.
Option C: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
This polynomial has one real root, [tex]\(-3\)[/tex]. Just like option B, it has exactly one real root and would also meet the required condition.
Option D: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
This polynomial has one real root, [tex]\(-3\)[/tex], and similarly meets the condition of having exactly one real root.
After analyzing the options, we find that Option B, which is [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex], has exactly one real root. This matches what we were searching for:
So, the correct answer is Option B.
Let's go through the polynomials one by one:
Option A: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
This polynomial has been found to have two real roots: [tex]\(-3\)[/tex] and [tex]\(3\)[/tex]. Therefore, it does not have exactly one real root.
Option B: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
This polynomial has exactly one real root: [tex]\(3\)[/tex]. Hence, this is a potential answer as it meets the condition of having exactly one real root.
Option C: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
This polynomial has one real root, [tex]\(-3\)[/tex]. Just like option B, it has exactly one real root and would also meet the required condition.
Option D: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
This polynomial has one real root, [tex]\(-3\)[/tex], and similarly meets the condition of having exactly one real root.
After analyzing the options, we find that Option B, which is [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex], has exactly one real root. This matches what we were searching for:
So, the correct answer is Option B.
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