0
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.
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