Conjugate Root Theorem
The Conjugate Root Theorem states that if is a polynomial with real coefficients, and is a root of the equation , where , then is also a root. A similar theorem states that if is a polynomial with rational coefficients and is a root of the polynomial, then is also a root.
Suppose that . Then . However, we know that , where we define to be the polynomial with the coefficients replaced with their complex conjugates; we know that by the assumption that has real coefficients. Thusly, we show that , and we are done.
This has many uses. If you get a fourth degree polynomial, and you are given that a number in the form of is a root, then you know that in the root. Using the Factor Theorem, you know that is also a root. Thus, you can multiply that out, and divide it by the original polynomial, to get a depressed quadratic equation. Of course, it doesn't have to be a fourth degree polynomial. It could just simplify it a bit.
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