solve polynomial equation by factoring

2x^3 - 12x^2 = 32x - 192

x^3-6x^2-16x-96=0

x^2(x-6)-16(x-6)=-

(x-6)(x+4)(x-4)=0

To solve the given polynomial equation by factoring, we need to rearrange the equation in standard form where one side of the equation is equal to zero. Let's do that:

2x^3 - 12x^2 - 32x + 192 = 0

Now, let's look for any common factors among the terms. In this case, we can factor out 2 from each term:

2(x^3 - 6x^2 - 16x + 96) = 0

Now, we can try to factor the remaining expression inside the parentheses. Since the power of the polynomial is 3, we can start by looking for a linear factor, which would be of the form (x - a).

We need to find a value of a such that when we substitute it into the expression, it will evaluate to zero.

By trying different values for a, we find that a = 4 makes the expression evaluate to zero. Therefore, (x - 4) is a factor of the polynomial.

Now we can use synthetic division or long division to divide the polynomial by (x - 4):

(x - 4) | x^3 - 6x^2 - 16x + 96

By performing the division, we get:

(x - 4)(x^2 - 2x - 24) = 0

Now we can solve for x by setting each factor equal to zero:

x - 4 = 0 or x^2 - 2x - 24 = 0

Solving the first equation, we get:

x = 4

For the second equation, we can factor it or use the quadratic formula. Factoring, we get:

(x - 6)(x + 4) = 0

Setting each factor equal to zero, we get:

x - 6 = 0 or x + 4 = 0

Solving these equations, we get:

x = 6 or x = -4

Therefore, the solutions to the given polynomial equation are x = 4, x = 6, and x = -4.