Solve x^3+6x^2+12x-117=0?

Question

x^3+6x^2+12x-117=0?

Answer 1

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

 x³ + 6x² + 12x - 117 = 0

If p/q is a rational zero, then p is a factor of 117 and q is a factor of 1.

The possible values of p are   ± 1, ± 3, ± 9, ± 13, ± 39 and   ± 117.

The possible values for q are ± 1.

So, p/q = ± 1, ± 3, ± 9, ± 13, ± 39 and ± 117.

Make a table for the synthetic division and test possible  zeros.

p/q	1	6	12	- 117
1	1	7	19	-98
-1	1	5	7	-124
3	1	9	39	0

Since f(3) = 0,   x = 3 is a zero. The depressed polynomial is   x² + 9x + 39 = 0

Since the depressed polynomial of this zero, x²  + 9x + 39, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

Roots are

The equation  x³ + 6x² + 12x - 117 = 0 have one real solution x = 3

and two imaginary soltions are .