Answer :
To find the zero of the function [tex]\( f(x) = 27x^3 - 27x^2 + 9x - 1 \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex].
Step 1: Set the function equal to zero.
[tex]\[ 27x^3 - 27x^2 + 9x - 1 = 0 \][/tex]
Step 2: Solve the equation for [tex]\( x \)[/tex].
Typically, solving a cubic equation can involve methods like factoring by grouping, synthetic division, or using the Rational Root Theorem to test potential rational roots. For this specific equation, you evaluate potential rational roots, simplify, and possibly use the quadratic formula if needed for the polynomial that remains.
After solving, the zero (root) of the function is:
[tex]\[ x = \frac{1}{3} \][/tex]
Thus, [tex]\( x = \frac{1}{3} \)[/tex] is the value where the function [tex]\( f(x) \)[/tex] equals zero.
Step 1: Set the function equal to zero.
[tex]\[ 27x^3 - 27x^2 + 9x - 1 = 0 \][/tex]
Step 2: Solve the equation for [tex]\( x \)[/tex].
Typically, solving a cubic equation can involve methods like factoring by grouping, synthetic division, or using the Rational Root Theorem to test potential rational roots. For this specific equation, you evaluate potential rational roots, simplify, and possibly use the quadratic formula if needed for the polynomial that remains.
After solving, the zero (root) of the function is:
[tex]\[ x = \frac{1}{3} \][/tex]
Thus, [tex]\( x = \frac{1}{3} \)[/tex] is the value where the function [tex]\( f(x) \)[/tex] equals zero.
