SOLUTION
We want to write the polynomial in factored form
[tex]x^3-4x^2-21x[/tex]
Looking at the last term of the polynomial -21x, we can take a smart guess that one of the zeros or roots of the polynomial would be -3. Now let us put x = -3 into the polynomial, if we get 0, then (x + 3) would be one of its factors, we have
[tex]\begin{gathered} x^3-4x^2-21x \\ (-3)^3-4(-3)^2-21(-3) \\ -27-4(9)+63 \\ =-27-36+63 \\ =0 \end{gathered}[/tex]
Hence (x + 3) is a factor. Now, dividing the polynomial with (x + 3), we have
[tex]\frac{x^3-4x^2-21x}{x+3}[/tex]
Dividing, we have
[tex]x^2-7x[/tex]
Factoring the polynomial we just got, we have
[tex]\begin{gathered} x^2-7x \\ x(x-7) \\ x=0,\text{ or } \\ x=7 \end{gathered}[/tex]
Hence the answer becomes
[tex]x(x+3)(x-7)[/tex]
