2.5 | Zeros of Polynomial Functions


The Fundamental Theorem of Algebra If \(f(x)\) is a polynomial of degree \(n,\) where \(n >0\), then \(f\) has at least one zero in the complex number system.
Linear Factorization Theorem If \(f(x)\) is a polynomial of degree \(n,\) where \(n >0,\) then \(f\) has precisely \(n\) linear factors
$$
f(x) = a_n(x -c_1)(x-c_2) \cdots (x-c_n)
$$
where \(c_1, c_2, \ldots, c_n\) are complex numbers.
The Rational Root Test
If the polynomial
$$
f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0
$$
has integer coefficients, every rational zero of \(f\) has the form
$$
\text{Rational zero} = \frac{p}{q}
$$
where \(p\) and \(q\) have no common factors other than \(1\), and

  • \(p = \) a factor of a constant term \(a_0\)
  • \(q = \) a factor of the leading coefficient \(a_n\)
Let \(f(x)\) be a polynomial that has real coefficients. If \(a+bi,\) where \(b \not = 0,\) is a zero fo the function, the conjugate \(a-bi\) is also a zero of the function.
Every polynomial of degree \(n>0\) with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.
Decartes’s Rule of Signs Let
$$
f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0
$$
be a polynomial with real coefficients and \(a_0 \not = 0.\)

  1. The number of positive real zeros of \(f\) is either equal to the number of variations in sign of \(f(x)\) or less than the number by an even integer.
  2. The number of negative real zeros of \(f\) is either equal to the number of variations in sign of \(f(-x)\) or less than that number by an even integer.
Let \(f(x)\) be a polynomial with real coefficients and a positive leading coefficient. Suppose \(f(x)\) is divided by \(x-c,\) using synthetic division.

  1. If \(c > 0\) and each number in the last row is either positive or zero, \(c\) is an upper bound for the real zero of \(f\).
  2. If \(c < 0\) and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), \(c\) is a lower bound for the real zeros of \(f.\)

E 2.5 Exercises