To find the roots of the polynomial equation:
[tex]\[
\frac{1}{2} x(x-7)(x+9)=0
\][/tex]
we need to set each factor of the equation equal to zero and solve for \( x \). This can be done step by step as follows:
1. The first factor is \( \frac{1}{2} x \). Setting this equal to zero:
[tex]\[
\frac{1}{2} x = 0
\][/tex]
Solving for \( x \):
[tex]\[
x = 0
\][/tex]
2. The second factor is \( x - 7 \). Setting this equal to zero:
[tex]\[
x - 7 = 0
\][/tex]
Solving for \( x \):
[tex]\[
x = 7
\][/tex]
3. The third factor is \( x + 9 \). Setting this equal to zero:
[tex]\[
x + 9 = 0
\][/tex]
Solving for \( x \):
[tex]\[
x = -9
\][/tex]
Therefore, the roots of the polynomial equation \( \frac{1}{2} x(x-7)(x+9)=0 \) are:
[tex]\[
-9, 0, 7
\][/tex]
So the correct answers are:
[tex]\[
-9, 0, 7
\][/tex]
