Can someone explain to how find the real roots of x3 plus 6x2-5x-30 equals 0?

Answer 1

x = 2.236, -2.236 & -6. (to 3 decimal places)

x3 + 6x2 - 5x - 30 = 0

First of all, let us note that as the highest power in the polynomial is to the power of 3, we are expecting to find 3 roots (the real + imaginary roots will equal 3).

Ok, so we need to factor the expression x3 + 6x2 - 5x - 30.

There are different methods to try to factor an expression like this. We are going to split this down into 2 groups (because we can see that we will be able to factor each).

1.x3 + 6x2 &

2. -5x - 30. We can factor out the "-" (minus) here to become: - (5x + 30).

So we have:

(x3 + 6x2) - (5x + 30) = 0

We can factor out "x2" from the first bracket and "5" from the second to get:

x2 (x + 6) -5 (x + 6) = 0

We can now see that "x + 6" is a common factor and so we can rewrite the original expression as:

(x2 - 5) (x + 6) = 0

Now we can solve for the roots of x by making each bracket equal to 0 in turn, so:

x2 - 5 = 0

x2 = 5

x = 51/2 (we have two roots here of course: the square root of 5 & the square root of 5 times -1)

x = 2.236 & -2.236 (to 3 decimal places only)

&

x + 6 = 0

x = -6

Thus all roots are real (none are imaginary) and they are: x = 2.236, -2.236 & -6.