Solve for x: 0 = 4x^3 - 24x^2 + 12x + 40

Steps to Solve

1. Simplify the equation

Divide the entire equation by 4 to simplify it: $$ \frac{4x^3}{4} - \frac{24x^2}{4} + \frac{12x}{4} + \frac{40}{4} = \frac{0}{4} $$ This simplifies to: $$ x^3 - 6x^2 + 3x + 10 = 0 $$

2. Find a root by testing factors of the constant term

The constant term is 10. We need to find a factor of 10 that, when substituted for $x$, makes the equation equal to zero. Factors of 10 include $\pm1, \pm2, \pm5, \pm10$. Test $x = -1$: $$ (-1)^3 - 6(-1)^2 + 3(-1) + 10 = -1 - 6 - 3 + 10 = 0$$ So, $x = -1$ is a root.

3. Perform synthetic division

Since $x = -1$ is a root, $(x + 1)$ is a factor. Use synthetic division to divide the polynomial $x^3 - 6x^2 + 3x + 10$ by $(x + 1)$.

-1 |  1  -6   3   10
    |     -1   7  -10
    -------------------
      1  -7  10    0

This gives us the quotient $x^2 - 7x + 10$.

4. Factor the quadratic equation

Now, factor the quadratic equation $x^2 - 7x + 10 = 0$. We are looking for two numbers that multiply to 10 and add up to -7. These numbers are -2 and -5. So, the quadratic equation factors to: $$ (x - 2)(x - 5) = 0$$

5. Solve for the remaining roots

Set each factor equal to zero and solve for $x$: $$ x - 2 = 0 \implies x = 2$$ $$ x - 5 = 0 \implies x = 5$$

6. List all roots

The roots of the original equation are $x = -1, 2, 5$.