To solve the equation [tex]\( 4z(8+z)(-7z-1) = 0 \)[/tex], we need to find the values of [tex]\( z \)[/tex] that make the equation true. Here’s a detailed step-by-step solution:
1. Understand the Equation:
The given equation is a product of three factors set equal to zero:
[tex]\[
4z(8+z)(-7z-1) = 0
\][/tex]
For the product to be zero, at least one of the factors must be zero.
2. Set Each Factor to Zero:
We will set each factor to zero and solve for [tex]\( z \)[/tex]:
- First factor:
[tex]\[
4z = 0
\][/tex]
Divide both sides by 4:
[tex]\[
z = 0
\][/tex]
- Second factor:
[tex]\[
8 + z = 0
\][/tex]
Solve for [tex]\( z \)[/tex]:
[tex]\[
z = -8
\][/tex]
- Third factor:
[tex]\[
-7z - 1 = 0
\][/tex]
Add 1 to both sides:
[tex]\[
-7z = 1
\][/tex]
Divide both sides by -7:
[tex]\[
z = -\frac{1}{7}
\][/tex]
3. List All Solutions:
The solutions to the equation [tex]\( 4z(8+z)(-7z-1) = 0 \)[/tex] are the values of [tex]\( z \)[/tex] that make any of the factors zero:
[tex]\[
z = 0, \quad z = -8, \quad z = -\frac{1}{7}
\][/tex]
Therefore, the final solutions are:
[tex]\[
z = 0, -8, -\frac{1}{7}
\][/tex]
