To find the real zeros of the function [tex]\( f(x) = 20x^3 - 16x^2 - 5x + 4 \)[/tex], we need to solve for [tex]\( x \)[/tex] where the function is equal to zero, i.e., [tex]\( f(x) = 0 \)[/tex].
Step-by-Step Solution:
1. Set up the equation:
[tex]\[
20x^3 - 16x^2 - 5x + 4 = 0
\][/tex]
2. Solve the cubic equation:
The cubic equation is of the form [tex]\( a x^3 + b x^2 + c x + d = 0 \)[/tex] where [tex]\( a = 20 \)[/tex], [tex]\( b = -16 \)[/tex], [tex]\( c = -5 \)[/tex], and [tex]\( d = 4 \)[/tex].
3. Determine the roots:
Solving cubic equations analytically involves finding the roots, which can be complex or real. For our function, the solutions are calculated and found to be:
[tex]\[
x = -0.500000000000000, \quad x = 0.500000000000000, \quad x = 0.800000000000000
\][/tex]
4. Verify if these roots are real:
Each of these values is real and can be verified by checking if substituting them back into the original equation results in a value sufficiently close to zero.
Therefore, the real zeros of the function [tex]\( f(x) = 20x^3 - 16x^2 - 5x + 4 \)[/tex] are:
[tex]\[
\boxed{-0.500000000000000, 0.500000000000000, 0.800000000000000}
\][/tex]
