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Find the solutions to this polynomial equation:

[tex]\[ 3(x-2)(x^2-9)(x+7)=0 \][/tex]

Select all the correct answers.

A. 2

B. -3

C. 3

D. -3

E. 7

F. -7



Answer :

GinnyAnswer
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]

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