As noted above, the purpose of using algebra in physics is so that we
can take known expressions that e.g. describe laws of nature and a
particular problem and transform these ``truths'' into a ``true''
statement of the answer by isolating the *symbol* for that answer on
one side of an equation.

For linear problems that is usually either straightforward or
impossible. For ``simple'' linear problems (a single linear equation)
it is always possible and usually easy. For sets of simultaneous linear
equations in a small number of variables (like the ones represented in
the course) one can ``always'' use a mix of composition (substitution)
and elimination to find the answer desired^{16}.

What about solving polynomials of higher degree to find values of their
variables that represent answers to physics (or other) questions? In
general one tries to arrange the polynomial into a *standard form*
like the one above, and then finds the *roots* of the polynomial.
How easy or difficult this may be depends on many things. In the case
of a *quadratic* (second degree polynomial involving at most the
square) one can - and we will, below - derive an *algebraic
expression* for the roots of an *arbitrary* quadratic.

For third and higher degrees, our ability to solve for the roots is not
trivially general. Sometimes we will be able to ``see'' how to go about
it. Other times we won't. There exist computational methodologies that
work for most relatively low degree polynomials but for very high degree
general polynomials the problem of factorization (finding the roots) is
*hard*. We will therefore work through quadratic forms in detail
below and then make a couple of observations that will help us factor a
few e.g. cubic or quartic polynomials should we encounter ones with one
of the ``easy'' forms.

In physics, quadratic forms are quite common. Motion in one dimension with constant acceleration (for example) quite often requires the solution of a quadratic in time. For the purposes of deriving the quadratic formula, we begin with the ``standard form'' of a quadratic equation:

(60) |

(where you should note well that , , in the general polynomial formula given above).

We wish to find the (two) values of
such that this equation is true,
given
. To do so we must rearrange this equation and *complete the square*.

0 | |||

(61) |

This last result is the well-known *quadratic formula* and its
general solutions are *complex numbers* (because the argument of the
square root can easily be negative if
). In some cases the
complex solution is *desired* as it leads one to e.g. a *complex
exponential* solution and hence a trigonometric oscillatory function as
we shall see in the next section. In other cases we insist on the
solution being real, because if it isn't there is no real solution to
the problem posed! Experience solving problems of both types is needed
so that a student can learn to recognize both situations and use complex
numbers to their advantage.

Before we move on, let us note two cases where we can ``easily'' solve
cubic or quartic polynomials (or higher order polynomials) for their
roots algebraically. One is when we take the quadratic formula and
multiply it by any power of
, so that it can be *factored*, e.g.

0 | |||

0 | (62) |

This equation clearly has the two quadratic roots given above plus one (or more, if the power of is higher) root . In some cases one can factor a solvable term of the form by inspection, but this is generally not easy if it is possible at all without solving for the roots some other way first.

The other "tricky" case follows from the observation that:

(63) |

so that the two roots are solutions. We can generalize this and solve e.g.

(64) |

and find the

In *this* course we will almost never have a problem that cannot be
solved using ``just'' the quadratic formula, perhaps augmented by one or
the other of these two tricks, although naturally a diligent and
motivated student contemplating a math or physics major will prepare for
the more difficult future by reviewing the various factorization tricks
for ``fortunate'' integer coefficient polynomials, such as *synthetic division*. However, such a student should *also* be aware
that the general problem of finding all the roots of a polynomial of
arbitrary degree is *difficult*^{17} . So
difficult, in fact, that it is known that no *simple* solution
involving only arithmetical operations and square roots exists for
degree 5 or greater. However it is generally fairly easy to factor
arbitrary polynomials to a high degree of accuracy *numerically*
using well-known algorithms and a computer.

Now that we understand both inverse functions and Taylor series
expansions and quadratics and roots, let us return to the question asked
earlier. What happens if we extend the domain of an inverse function
outside of the range of the original function? In general we find that
the inverse function has no *real* solutions. Or, we can find as
noted above when factoring polynomials that like as not there are no
real solutions. But that does not mean that solutions do not exist!