To identify the zeros of the function [tex]\( f(x) = (x-7)(x+4)(3x-2) \)[/tex], we need to set the function equal to zero and solve for [tex]\( x \)[/tex].
A function equals zero when any of its factors equal zero. The factors of [tex]\( f(x) \)[/tex] are [tex]\((x-7)\)[/tex], [tex]\((x+4)\)[/tex], and [tex]\((3x-2)\)[/tex].
We solve for [tex]\( x \)[/tex] in each of these factors:
1. First factor:
[tex]\[
x - 7 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 7
\][/tex]
2. Second factor:
[tex]\[
x + 4 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -4
\][/tex]
3. Third factor:
[tex]\[
3x - 2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
3x = 2 \implies x = \frac{2}{3}
\][/tex]
The zeros of the function [tex]\( f(x) = (x-7)(x+4)(3x-2) \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = -4 \)[/tex], and [tex]\( x = \frac{2}{3} \)[/tex].
Therefore, the correct answer from the given choices is:
[tex]\[
-4, \frac{2}{3}, 7
\][/tex]
