Write the polynomial in factored form. Please show your work or explain your reasoning
x^3 - 5x^2 + 6x
To factor the polynomial x^3 - 5x^2 + 6x, we can first look for a common factor among the terms. In this case, the only common factor is x. Factoring out x from each term gives us:
x(x^2 - 5x + 6)
Now we need to factor the quadratic expression x^2 - 5x + 6. To do this, we can look for two numbers whose product is 6 and whose sum is -5 (the coefficient of the x term). The numbers -2 and -3 satisfy these conditions because -2 * -3 = 6 and -2 + (-3) = -5.
Therefore, we can rewrite the quadratic expression as:
x(x - 2)(x - 3)
So the factored form of the polynomial x^3 - 5x^2 + 6x is x(x - 2)(x - 3).