find all roots: x^3 + 7x^2 + 12x = 0
show all work and check your answer.

The three roots of x^3 + 7x^2 + 12x = 0 is 0,-3 and -4

Solution:

We have been given a cubic polynomial.

We need to find the three roots of the given polynomial.

Since it is a cubic polynomial, we can start by taking ‘x’ common from the equation.

This gives us:

  ----- eqn 1

So, from the above eq1 we can find the first root of the polynomial, which will be:

x = 0

Now, we need to find the remaining two roots which are taken from the remaining part of the equation which is:

we have to use the quadratic equation to solve this polynomial. The quadratic formula is:

Now, a = 1, b = 7 and c = 12

By substituting the values of a,b and c in the quadratic equation we get;

Therefore, the two roots are:

And,

Hence, the three roots of the given cubic polynomial is 0, -3 and -4