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Answer :
To determine which of the given polynomials is a 3rd degree polynomial with exactly one real root, let's analyze each option.
A polynomial of degree 3 can have up to 3 real roots, and we want to identify the polynomial that has exactly 1 real root. Checking each polynomial for the number of real roots and selecting the one that meets the criteria is the approach we will use.
Let's consider each polynomial option one by one:
Option A: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
This polynomial needs to be analyzed to check how many real roots it has. Based on calculations, it turns out that this polynomial has exactly 1 real root.
Option B: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
This polynomial might have 1 or more real roots, but in our checks, it reveals different results that don't meet the condition of having exactly 1 real root.
Option C: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
Similarly, this polynomial also provides results that suggest it does not have exactly 1 real root.
Option D: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
Like options B and C, checking the number of real roots for this polynomial shows results inconsistent with having exactly 1 real root.
After reviewing all the possibilities, it turns out that Option A with [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex] is the only polynomial that has exactly 1 real root. Hence, the answer is:
A. [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
A polynomial of degree 3 can have up to 3 real roots, and we want to identify the polynomial that has exactly 1 real root. Checking each polynomial for the number of real roots and selecting the one that meets the criteria is the approach we will use.
Let's consider each polynomial option one by one:
Option A: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
This polynomial needs to be analyzed to check how many real roots it has. Based on calculations, it turns out that this polynomial has exactly 1 real root.
Option B: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
This polynomial might have 1 or more real roots, but in our checks, it reveals different results that don't meet the condition of having exactly 1 real root.
Option C: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
Similarly, this polynomial also provides results that suggest it does not have exactly 1 real root.
Option D: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
Like options B and C, checking the number of real roots for this polynomial shows results inconsistent with having exactly 1 real root.
After reviewing all the possibilities, it turns out that Option A with [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex] is the only polynomial that has exactly 1 real root. Hence, the answer is:
A. [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
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