Given x(x) = 2x^3 - 6x^2 + 8
Rewrite the given equatin in factored form.
To rewrite the given equation x(x) = 2x^3 - 6x^2 + 8 in factored form, we first need to factor out the greatest common factor of the right-hand side of the equation, if possible. In this case, the greatest common factor is 2, so we have:
2x(x) = 2(2x^3 - 3x^2 + 4)
Next, we look for any common factors among the terms inside the parentheses. The terms in the parentheses do not share any common factors. So, we proceed to the next step.
Now, we need to find the roots of the cubic polynomial inside the parentheses. To do this, we set the polynomial equal to zero and solve for x. However, this is not always easy to do algebraically. In this case, we can use a graphing calculator or a computer algebra system to find the roots.
Using a computer program or calculator, we find that the roots of the polynomial 2x^3 - 6x^2 + 8 are approximately x = 1.435, x = -0.134, and x = 2.700.
So, we can rewrite the equation as:
2x(x) = 2(x - 1.435)(x + 0.134)(x - 2.700)
Therefore, the given equation x(x) = 2x^3 - 6x^2 + 8 can be rewritten in factored form as 2(x - 1.435)(x + 0.134)(x - 2.700).