Complex solution of a cubic function.

In summary, the conversation discusses solving a cubic equation with complex numbers. The person was able to find two real solutions but struggled to find the complex one. The expert suggests dividing the equation by z+2 and z-3 to factor it into linear terms and find the solution.
  • #1
yoyoyo992
2
0
Hi,
I'm studying my algebra and I was trying this exercise:

Solve
z³+iz²-7z-iz-6-6i=0

I found the 2 real solutions (3 and -2) but i can't seem to find the complex one.

I tried this:

(z²-z)(i)=6(i) ==> z= -2 and 3
z³-7z=6==> z= -1, -2 and 3

I found that the complex solution is -1-i but i have no idea how to find it.

(Im from Belgium, it's difficult to type maths in another language so sorry for mistakes)
 
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  • #2
Well, the obvious thing to do is divide by z+ 2 and z- 3 so that you can factor that cubic into linear terms, then set each term equal to 0.
 
  • #3
Thanks, I found it
 

Related to Complex solution of a cubic function.

What is a cubic function?

A cubic function is a type of polynomial function with a degree of three. It can be written in the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.

What is a complex solution?

A complex solution is a solution to an equation that involves imaginary numbers. In the case of a cubic function, a complex solution would involve taking the square root of a negative number, which is represented by the letter i.

How do you find the complex solutions of a cubic function?

To find the complex solutions of a cubic function, you can use the quadratic formula to solve for the values of x. The expression under the square root sign will result in a negative number, indicating a complex solution. You can then use the imaginary unit i to represent the square root of the negative number.

Why do complex solutions occur in cubic functions?

Complex solutions occur in cubic functions because of the nature of the cubic equation. When solving for the roots of a cubic function, it is possible to get a negative under the square root sign, which results in a complex solution. This is due to the fact that cubic functions can have up to three distinct roots.

What is the significance of complex solutions in real-world applications?

In real-world applications, complex solutions play a crucial role in solving problems related to physics, engineering, and other fields. For example, in electrical engineering, complex solutions are used to analyze circuits with AC current. They also have applications in solving certain differential equations and in signal processing.

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