Finding the roots of higher degree polynomials is much more
difficult than finding the roots of a quadratic function. A few tools do
make it easier, though. 1) If *r* is a root of a polynomial function, then
(*x* - *r*) is a factor of the polynomial. 2) Any polynomial with real
coefficients can be written as the product of linear factors (of the form (*x* - *r*)) and quadratic factors which are irreducible over the real numbers. A
quadratic factor that is irreducible over the reals is a quadratic function with
no real solutions; that is, *b*^{2} -4*ac* < 0. All factors, linear and quadratic,
will have real coefficients.

Two other theorems also have to do with the roots of a polynomial, Descartes'
Rule of Signs, and the Rational Root Theorem.

Descartes' Rule of Signs has to do with the number of real roots possible for a
given polynomial function *f* (*x*). The number of variations in a polynomial
is the number of times two consecutive terms of the polynomial (*a*_{2}*x*^{2} and
*a*_{1}*x* for example) have different signs. Descartes' Rule of Signs states that
the number of positive real roots is less than or equal to the number of
variations in the function *f* (*x*). It also states that the number of negative
real roots is less than or equal to the number of variations in the function
*f* (- *x*). Furthermore, in either case, the difference between the number of
variations and the number of real roots will always be an even integer.

The Rational Root Theorem is another useful tool in finding the roots of a
polynomial function *f* (*x*) = *a*_{n}*x*^{n} + *a*_{n-1}*x*^{n-1} + ... + *a*_{2}*x*^{2} + *a*_{1}*x* + *a*_{0}.
If the coefficients of a polynomial are all integers, and a root of the
polynomial is rational (it can be expressed as a fraction in lowest terms), the
numerator of the root is a factor of *a*_{0} and the denominator of the root is a
factor of *a*_{n}.

Using these tools, let's examine a sample polynomial function: *p*(*x*) = *x*^{4} +4*x*^{3} -8*x*^{2} - 33*x* - 18. There is one variation in *p*(*x*), so the number of
positive roots is one. *p*(- *x*) = *x*^{4} -4*x*^{3} -7*x*^{2} + 33*x* - 18. *p*(- *x*) has
three variations, so there are either three or one negative roots (there cannot
be two because then the difference between variations and roots would not be an
even integer).

Next we can use the Rational Root Theorem to look for any rational roots. The
factors of *a*_{0} = - 18 are ±1, ±2, ±3, ±6, ±9, ±18. The factors of *a*_{n} = 1 are ±1. Therefore the possible rational
roots are ±1, ±2, ±3, ±6, ±9, and ±18. Checking
each of these possibilities using synthetic division, we find that the only
rational roots are *x* = -2, 3. We can now divide the polynomial by (*x* + 2)(*x* - 3) to arrive at the quotient (*x*^{2} + 5*x* + 3). If this quotient were
constant, then we would have found all of the roots of the polynomial. As it
is, the quotient is a quadratic function. If it has real roots, they are
irrational. It may have no real roots, in which case we are done. Using the
quadratic formula, we find the real roots of the quadratic factor are - 0.69 and - 4.30. So indeed there are three negative roots, and one
positive root, but only two rational roots. All in all there are four real
roots.

In other situations, there may be no variations in a function, in which
potential roots either greater than or less than zero can be eliminated from the
possibilities. In other circumstances, a quadratic factor is irreducible over
the real numbers, and has only complex roots. There are also situations in
which the same root factors into the polynomial twice. Although the graph of
such a polynomial crosses the *x*-axis at that root only once, the root is
counted twice. It is said to have multiplicity of two. Whenever (*x* - *r*)^{m} is a factor of a polynomial, but (*x* - *r*)^{(}*m* + 1) is not, then that root,
*r*, is a root of multiplicity *m*.

Complex roots will not be discussed
until after a thorough
exploration of
complex numbers and polar
coordinates. Complex
numbers are an important part of finding the roots
of a polynomial, though. When a quadratic function is irreducible over the real
numbers, complex roots exist. The Fundamental Theorem of Algebra states that
every polynomial has at least one complex root. Furthermore, it can be proved
that, including complex roots and each multiplicity counted as a different root,
a polynomial with degree *n* always has exactly *n* roots. At this point,
though, we'll concern ourselves exclusively with finding real roots.