Answer:
[tex]x=0,\:x=-3,\:x=-3,\:x=2i,\:x=-2i[/tex]
Step-by-step explanation:
[tex]g(x)=x^5+6x^4+13x^3+24x^2+36x = x (x^4+6x^3+13x^2+24x+36)[/tex]
We have a zero at x = 0.
Possible rational zeros: (highest and lowest power's coefficients)
36 ⇒ 1, 2, 3, 4, 6, 9, 12, 18, 36
1 ⇒ 1
+ {1, 2, 3, 4, 6, 9, 12, 18, 36} and - {1, 2, 3, 4, 6, 9, 12, 18, 36}
f(-3) = 81 - 162 + 117 - 72 + 36 = 0
So, (x+3) is a factor.
Here comes the synthetic division part that I have shown in the attachment.
As shown in the attachment the function has double zeros at x = -3.
After synthetic division, the remaining part is [tex](x^2+4)[/tex]
It also means that we have 2 complex zeros at x = 2i and x=-2i
So the final form of g(x) is [tex]g(x) = x(x+3)^2(x^2+4)[/tex]
Final solution is [tex]x=0,\:x=-3,\:x=-3,\:x=2i,\:x=-2i[/tex]
