Factoring and Expanding: Unveiling the Relationship Between Complex Numbers and Polynomial Equations
This article delves into the intriguing relationship between complex numbers and polynomial equations, specifically focusing on the equation (x+1+i)(x+1i)(x1+i)(x1i) = x^4 + 4.
Understanding the Complex Conjugates
The equation features complex numbers in the form of a + bi, where a and b are real numbers and i is the imaginary unit (√1). Notice that each pair of factors in the equation consists of complex conjugates, meaning they differ only in the sign of their imaginary part. For example, (x + 1 + i) and (x + 1  i) are complex conjugates.
The Key Property: Product of Complex Conjugates
A crucial property of complex conjugates is that their product results in a real number. This is because:
(a + bi)(a  bi) = a²  (bi)² = a² + b²
This property will play a vital role in simplifying the equation.
Expanding the Equation
Let's expand the left side of the equation step by step using the property of complex conjugates:

(x+1+i)(x+1i) = (x+1)²  i² = x² + 2x + 1 + 1 = x² + 2x + 2

(x1+i)(x1i) = (x1)²  i² = x²  2x + 1 + 1 = x²  2x + 2
Now, multiplying the results from steps 1 and 2:
 (x² + 2x + 2)(x²  2x + 2) = (x² + 2)²  (2x)² = x⁴ + 4x² + 4  4x² = x⁴ + 4
Conclusion
As we have shown through expansion, the equation (x+1+i)(x+1i)(x1+i)(x1i) = x⁴ + 4 is indeed true. This example beautifully illustrates how complex conjugates simplify calculations and reveal the connection between complex numbers and polynomial equations. The equation can be expressed as a factored form, showcasing how complex numbers can be used to factor polynomials that are otherwise difficult to factor using real numbers only.