To solve the polynomial equation [tex]\(x^3 - x^2 - 6x = 0\)[/tex], we'll follow a step-by-step approach:
1. Factor out the common term:
Notice that each term in the polynomial has a common factor of [tex]\(x\)[/tex]. So, we can factor [tex]\(x\)[/tex] out of the equation:
[tex]\[ x^3 - x^2 - 6x = x(x^2 - x - 6) \][/tex]
2. Set each factor to zero:
For the product [tex]\(x(x^2 - x - 6)\)[/tex] to be zero, at least one of the factors must be zero:
[tex]\[ x = 0 \quad \text{or} \quad x^2 - x - 6 = 0 \][/tex]
3. Solve the quadratic equation:
Next, we need to solve the quadratic equation [tex]\(x^2 - x - 6 = 0\)[/tex]. To do this, we can factor the quadratic expression. We look for two numbers that multiply to [tex]\(-6\)[/tex] and add up to [tex]\(-1\)[/tex]:
[tex]\[ x^2 - x - 6 = (x - 3)(x + 2) \][/tex]
4. Set each factor of the quadratic equation to zero:
[tex]\[ x - 3 = 0 \quad \text{or} \quad x + 2 = 0 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
By solving each equation, we find:
[tex]\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \][/tex]
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
6. Combine all solutions:
The complete set of solutions from both steps are [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], and [tex]\(x = -2\)[/tex].
7. Order the solutions in ascending order:
Finally, to list the solutions in ascending order, we get:
[tex]\[ x = -2, 0, 3 \][/tex]
So, the solutions to the equation [tex]\(x^3 - x^2 - 6x = 0\)[/tex] are:
[tex]\[ x = -2, 0, 3 \][/tex]
