Find the zeros of the function below algebraically, if any exist.
$$f ( x ) = x ^ { 5 } - 9 x ^ { 3 } + 27 x ^ { 2 } - 243$$
A) $- 2$ and 2
B) $- 4$ and 2
C) $- 4$ and 4
D) $- 3$ and 4
E) $- 3$ and 3

Question

Quizplus · Accepted Answer

To find the zeros of the function $f(x) = x^5 - 9x^3 + 27x^2 - 243$, we look for values of $x$ that make $f(x) = 0$. Factoring by grouping or using synthetic division might be challenging due to the nature of the polynomial. However, by inspection or trial and error, we can check the given options. Plugging $x = -3$ and $x = 3$ into the function:For $x = -3$:$$f(-3) = (-3)^5 - 9(-3)^3 + 27(-3)^2 - 243 = -243 + 243 + 243 - 243 = 0$$For $x = 3$:$$f(3) = (3)^5 - 9(3)^3 + 27(3)^2 - 243 = 243 - 243 + 243 - 243 = 0$$Thus, $x = -3$ and $x = 3$ are zeros of the function, making option E correct.

Find the Zeros of the Function Below Algebraically, If Any f(x)=x5−9x3+27x2−243f ( x ) = x ^ { 5 } - 9 x ^ { 3 } + 27 x ^ { 2 } - 243f(x)=x5−9x3+27x2−243

Find the Zeros of the Function Below Algebraically, If Any $f ( x ) = x ^ { 5 } - 9 x ^ { 3 } + 27 x ^ { 2 } - 243$