Polynomial function theorems for zeros

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Polynomial function theorems for zeros are a set of theorems aiming to find (or determine the nature of) the polynomial remainder theorem:

Background[edit]

A polynomial function is a function of the form

 p(x) = a_n x^n + a_{n-1} x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0,

where  a_i\, (i = 0, 1, 2, \dotsc, n) are complex numbers and  a_n \ne 0 .

If  p(z) = a_n z^n + a_{n-1} z^{n-1} + \dotsb + a_2 z^2 + a_1 z + a_0 = 0, then z is called a zero of p(x). If z is real, then z is a real zero of p(x); if z is imaginary, the z is a complex zero of p(x), although complex zeros include both real and imaginary zeros.

The fundamental theorem of algebra states that every polynomial function of degree  n \ge 1 has at least one complex zero. It follows that every polynomial function of degree  n \ge 1 has exactly  n complex zeros, not necessarily distinct.

  • If the degree of the polynomial function is 1, i.e.,  p(x) = a_1 x + a_0, then its (only) zero is \frac{-a_0}{a_1}.
  • If the degree of the polynomial function is 2, i.e.,  p(x) = a_2 x^2 + a_1 x + a_0, then its two zeros (not necessarily distinct) are \frac{-a_1 + \sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} and \frac{-a_1 - \sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} .

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 theorems[edit]

Remainder theorem[edit]

The remainder theorem states that if p(x) is divided by x - c, then the remainder is p(c).

For example, when p(x) = x^3 + 2x - 3 is divided by x - 2, the remainder (if we do not care about the quotient) will be p(2) = 2^3 + 2(2) - 3 = 9. When p(x) is divided by x + 1, the remainder is p(-1) = (-1)^3 + 2(-1) - 3 = -6. However, this theorem is most useful when the remainder is 0 since it will yield a zero of p(x). For example, p(x) is divided by x - 1, the remainder is p(1) = (1)^3 + 2(1) - 3 = 0, so 1 is a zero of p(x) (by the definition of zero of a polynomial function).