Answer :
To determine which of the given polynomials is a 3rd degree polynomial with exactly 1 real root, we need to analyze each option.
A 3rd degree polynomial is also called a cubic polynomial and generally has three roots. These roots can be either real or complex. Our goal is to find which polynomial has exactly one real root.
Let's evaluate each option:
A. [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
The polynomial is a cubic equation, and upon evaluating its roots, it has exactly 1 real root.
B. [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
Similarly, this polynomial is also a cubic equation. Upon examining its roots, it has exactly 1 real root.
C. [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
This polynomial is a cubic equation. After checking, it has exactly 1 real root.
D. [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
This is a cubic equation, and after evaluating its roots, it has 2 real roots.
So, the polynomials in options A, B, and C each have exactly 1 real root, while option D has more than 1 real root. Therefore, options A, B, and C are the 3rd degree polynomials with exactly 1 real root.
A 3rd degree polynomial is also called a cubic polynomial and generally has three roots. These roots can be either real or complex. Our goal is to find which polynomial has exactly one real root.
Let's evaluate each option:
A. [tex]\( F(x) = x^3 + 9x^2 + 27x + 27 \)[/tex]
The polynomial is a cubic equation, and upon evaluating its roots, it has exactly 1 real root.
B. [tex]\( F(x) = x^3 + 3x^2 + 9x + 27 \)[/tex]
Similarly, this polynomial is also a cubic equation. Upon examining its roots, it has exactly 1 real root.
C. [tex]\( F(x) = x^3 - 9x^2 + 27x - 27 \)[/tex]
This polynomial is a cubic equation. After checking, it has exactly 1 real root.
D. [tex]\( F(x) = x^3 + 3x^2 - 9x - 27 \)[/tex]
This is a cubic equation, and after evaluating its roots, it has 2 real roots.
So, the polynomials in options A, B, and C each have exactly 1 real root, while option D has more than 1 real root. Therefore, options A, B, and C are the 3rd degree polynomials with exactly 1 real root.
