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, we can simplify the equation to find its roots. The given polynomial does not factor nicely in a straightforward manner, but we can use the Rational Root Theorem or numerical methods to test the options provided. Testing the options given, we find that $x = -3$ and $x = 3$ satisfy the equation, making them the zeros of the function. This is because plugging $x = -3$ or $x = 3$ into the polynomial results in $0$, which means they are the roots of the equation.

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 $f ( x ) = x ^ { 5 } - 9 x ^ { 3 } + 27 x ^ { 2 } - 243$