Factor the polynomial completely
please show all work/steps
x^3-4x^2-21x
To factor the given polynomial, we can first look for the greatest common factor. In this case, we can see that each term has a factor of x, so we can factor out x:
x^3 - 4x^2 - 21x = x(x^2 - 4x - 21)
Next, we will try to factor the quadratic expression inside the parentheses: x^2 - 4x - 21.
To do this, we need to find two numbers that multiply to -21 and add to -4. The numbers -7 and 3 satisfy these conditions, so we can rewrite the quadratic as:
x^2 - 4x - 21 = x(x - 7) + 3(x - 7)
Now, we can factor out the common binomial factor of (x - 7):
x(x - 7) + 3(x - 7) = (x + 3)(x - 7)
Putting it all together, the polynomial x^3 - 4x^2 - 21x can be factored as:
x(x + 3)(x - 7)