Explain how a polynomial can have roots that are not complex conjugates

Question

Question: The roots of (x-1)(x+i) = 0 are obviously 1, -i. These are not complex conjugates. How do you explain this? Note that 'i' represents the imaginary number i.

Answer 1

If a polynomial has REAL coefficients, then complex roots occur in conjugate pairs.  A polynomial with IMAGINARY coefficients will have at least one imaginary root whose complex conjugate is not a root.  The example you give is such a polynomial.   

Answer 2

Your premise is incorrect: the roots of (x-1)(x+i)=0 are not obviously 1, -i

 

Allow me to demonstrate:

 

1. We know that i represents the imaginary square root of -1, and thus, that i² = -1

 

2. Next we are going to eliminate the imaginary component of this equation through substitution and squaring, after doing a little rearranging:

 

(x-1)(x+i) = 0

 

x²  + xi - x - i = 0

 

x² + xi - x = i

 

Now we multiply both sides by i to get:

 

x²i + xi² - xi = i²

 

Then substitute -1 for i² on both sides:

 

x²i - x - xi = -1

 

Some more rearranging to further isolate i:

 

x²i - xi = x - 1

 

i(x² - x) = x - 1

 

Now we deal a death blow to that final pesky i through squaring both sides and substituting -1 again for i²:

 

i²(x² - x)² = (x - 1)²

 

-(x² - x)^{2 =} x² - 2x + 1

 

Finally we reduce both sides of the equation in order to get a polynomial on one side and zero on the other:

 

-(x⁴ - 2x³ - x²) = x² - 2x + 1

 

-x⁴ + 2x³ + x² = x² - 2x +1

 

0 = x⁴ - 2x³ - 2x + 1

 

This looks very different from where we started but no established mathematical rules have been broken.  When you graph the final polynomial in an equation as P(x) = x⁴ - 2x³ - 2x + 1 you can clearly see where it intercepts the x-axis in two locations, meaning there are two real roots, and neither of them is equal to 1.  It is not so obvious what the roots are when imaginary numbers are involved, in fact I would suggest that no real information can ever be derived from any equation or expression involving an imaginary number.