Polynomial function theorems for zeros
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- Factor theorem
- Descartes' rule of signs
- Gauss–Lucas theorem
- Rational zeros theorem
- Bounds on zeros theorem also known as the boundedness theorem
- Intermediate value theorem
- Complex conjugate root theorem
- Properties of polynomial roots
where are complex numbers and .
If , then is called a zero of . If is real, then is a real zero of ; if is imaginary, the is a complex zero of , although complex zeros include both real and imaginary zeros.
The fundamental theorem of algebra states that every polynomial function of degree has at least one complex zero. It follows that every polynomial function of degree has exactly complex zeros, not necessarily distinct.
- If the degree of the polynomial function is 1, i.e., , then its (only) zero is .
- If the degree of the polynomial function is 2, i.e., , then its two zeros (not necessarily distinct) are and .
A degree one polynomial is also known as a linear function, whereas a degree two polynomial is also known as a quadratic function and its two zeros are merely a direct result of the quadratic formula. However, difficulty rises when the degree of the polynomial, n, is higher than 2. There is a cubic formula for a cubic function (a degree three polynomial) and there is a quartic formula for a quartic function (a degree four polynomial), but they are very complicated. There is no general formula for a polynomial function of degree 5 or higher (see Abel–Ruffini theorem).
The remainder theorem states that if is divided by , then the remainder is .
For example, when is divided by , the remainder (if we do not care about the quotient) will be . When is divided by , the remainder is . However, this theorem is most useful when the remainder is 0 since it will yield a zero of . For example, is divided by , the remainder is , so 1 is a zero of (by the definition of zero of a polynomial function).