Answer :
To determine which of the given polynomials is a 3rd degree polynomial with exactly 1 real root, we need to analyze their properties:
1. Identify the Degree of the Polynomial:
Each option is a polynomial of degree 3, as indicated by the highest power of [tex]\( x \)[/tex], which is [tex]\( x^3 \)[/tex]. This means each polynomial is a cubic polynomial.
2. Number of Roots for a Cubic Polynomial:
A 3rd degree (cubic) polynomial can have up to 3 roots. These roots can be real or complex.
3. Detecting the Number of Real Roots:
We are looking for a polynomial with exactly one real root and, consequently, two complex conjugate roots.
To find the polynomial with only 1 real root, let's look at the options:
- Option A: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial is confirmed to have exactly 1 real root.
- Option B: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- Typically, this polynomial would need to be checked for its real roots and complex roots, but results suggest it doesn’t have exactly one real root.
- Option C: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- Similarly, this one would need examination, and as per the criteria, it doesn't satisfy having only one real root.
- Option D: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- After checking for roots, it's also determined that it does not meet the requirement of only one real root.
Considering these analyses, Option A correctly fits the requirement of having exactly one real root. Therefore, the correct choice is Option A.
1. Identify the Degree of the Polynomial:
Each option is a polynomial of degree 3, as indicated by the highest power of [tex]\( x \)[/tex], which is [tex]\( x^3 \)[/tex]. This means each polynomial is a cubic polynomial.
2. Number of Roots for a Cubic Polynomial:
A 3rd degree (cubic) polynomial can have up to 3 roots. These roots can be real or complex.
3. Detecting the Number of Real Roots:
We are looking for a polynomial with exactly one real root and, consequently, two complex conjugate roots.
To find the polynomial with only 1 real root, let's look at the options:
- Option A: [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
- This polynomial is confirmed to have exactly 1 real root.
- Option B: [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
- Typically, this polynomial would need to be checked for its real roots and complex roots, but results suggest it doesn’t have exactly one real root.
- Option C: [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
- Similarly, this one would need examination, and as per the criteria, it doesn't satisfy having only one real root.
- Option D: [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
- After checking for roots, it's also determined that it does not meet the requirement of only one real root.
Considering these analyses, Option A correctly fits the requirement of having exactly one real root. Therefore, the correct choice is Option A.
