Can Substituting -1 for i in Complex Cubic Equations Yield Accurate Solutions?

In summary, James was trying to find the solutions to a polynomial with complex coefficients, but he was lucky that his dodgy maths worked.
  • #1
Noir
27
0
I know this isn't in the right format, but I figured I'd get a better answer here than anywhere else. In my last exam, there was a question asking to prove (a + bi - except there were values for a and b, but i forgot them) was a solution to a polynomial of the 3rd degree.

Said polynomial was complex. I sub'd it in and got 0, so it worked. Here's the catch. To find the other two solutions, I subbed -1 in for i so the equation wasn't complex. I have no idea why I did it, i just remember it working. Lo and behold I got it right, the cubic equation I got had the same solutions as the quadratic you would get if you found used the long division method. I wasn't thinking and didn't want to deal with the i's.

I want to know, does the above methord work? Subbing in -1 for i and then solving? If it does I just went from a B to an A and I'm happy. I tried looking on the internet, but couldn't find anything.

Cheers,

James.
 
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  • #2
Unfortunately, I have no idea what you are saying. Do you mean that the coefficients of the polynomial themselves were complex numbers? It would help a lot if you were to show what polynomial you are talking about. In general replacing "i" with "-1" will not give you anything worhwhile ("i" and "-1" are NOT equal and one CANNOT replace the other) so it sounds like you were just lucky in this particular example.
 
  • #3
Sorry for any confusion. Much like this equation right here;
z^4 + 3iz^3 - (4 + i)z^2 - 3iz + 3 + i = 0
I think your right; I was just lucky my dodgy maths worked! I don't see how replacing i with -1 would, or even could work now...
 

1. What is a polynomial?

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, which are combined using addition, subtraction, and multiplication. The highest exponent in a polynomial is known as the degree of the polynomial.

2. What is the difference between solving a polynomial over C and over R?

Solving a polynomial over C refers to finding all complex solutions, while solving over R refers to finding all real solutions. This is because the set of complex numbers (C) includes all real numbers (R) as well as imaginary numbers.

3. How do you solve a polynomial over C?

To solve a polynomial over C, you can use the fundamental theorem of algebra, which states that a polynomial of degree n has exactly n complex solutions. You can also use methods such as factoring, the quadratic formula, or the rational root theorem.

4. Why is solving polynomials over C important?

Solving polynomials over C is important because it allows us to find the roots or solutions of a polynomial, which in turn helps us understand the behavior and characteristics of the polynomial. This is useful in many applications, including engineering, physics, and economics.

5. Can polynomials over C have no solutions?

Yes, polynomials over C can have no solutions. This occurs when all the roots of the polynomial are complex numbers and there are no real solutions. However, the fundamental theorem of algebra guarantees that a polynomial of degree n will have exactly n complex solutions, counting multiplicities.

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