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.