Certainly! Let's solve the polynomial equation step-by-step:
Given polynomial equation:
[tex]\[
3(x-2)\left(x^2-9\right)(x+7) = 0
\][/tex]
To find the solutions, we need to set each factor equal to [tex]\(0\)[/tex] and solve for [tex]\(x\)[/tex].
### Step 1: Solving the First Factor
The first factor is:
[tex]\[
(x - 2) = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = 2
\][/tex]
### Step 2: Solving the Second Factor
The second factor is:
[tex]\[
(x^2 - 9) = 0
\][/tex]
This can be factored further using the difference of squares:
[tex]\[
(x - 3)(x + 3) = 0
\][/tex]
Now, set each factor equal to [tex]\(0\)[/tex] and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 3 = 0 \implies x = 3
\][/tex]
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
### Step 3: Solving the Third Factor
The third factor is:
[tex]\[
(x + 7) = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = -7
\][/tex]
### Conclusion
Putting it all together, the solutions to the polynomial equation
[tex]\[
3(x-2)\left(x^2-9\right)(x+7)=0
\][/tex]
are:
[tex]\[
x = 2, 3, -3, -7
\][/tex]
Therefore, the correct answers are:
[tex]\[
2, 3, -3, -7
\][/tex]
