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.
$$
f(x) = a_n(x -c_1)(x-c_2) \cdots (x-c_n)
$$
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
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 \(a_0 \not = 0.\)
$$
f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0
$$
be a polynomial with real coefficients and \(a_0 \not = 0.\)
- 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.
- 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.
- 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 \(f\).
- 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 \(f.\)