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 complex zeros of a polynomial function.

Found in most precalculus textbooks, these theorems include:

[edit] Background

A polynomial function is a function of the form

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

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

If p(z) = anzn + an − 1zn − 1 + ... + a2z2 + a1z + a0 = 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 ncomplex 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. It is true that 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 complicate. To make matter worst, there is no general formula for a polynomial function of degree 5 or higher.

[edit] The theorems

[edit] Remainder theorem

The remainder theorem states that if p(x) is divided by x − c, then the remainder is p(c).
For example, when p(x) = x3 + 2x − 3 is divided by x − 2, the remainder (if we don't care about the quotient) will be p(2) = 23 + 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).

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