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Factoring General Polynomials

Just as every number has a prime factorization (e.g. ) we can also factor a polynomial into the product of a number of simpler expressions of the form  where a and b are constants.  Determining a polynomial’s factorization is useful for several reasons.  Factoring makes it easy to find the roots of a polynomial.  It can often give us insights about the graph of the polynomial, and it is useful in solving polynomial equations.

The tutorial entitled Solving Quadratic Equations by Factoring, explains how to factor a quadratic expression, but this tutorial extends that idea further to include factoring higher degree polynomials.

Example

Completely factor the polynomial:

.

Solution

One approach to factoring is to first look for a root, k, of this polynomial (i.e. find a solution to the equation ).  Then by the discussion at the end of the tutorial entitled Long Division of Polynomials, we know that  is a factor of .  This is the approach we will take.

While there is a formula, like the quadratic formula, that will solve the equation , it is much more complicated than the quadratic formula, so instead we will check first to see if the rational root test can help us out (refer to the tutorial entitled Rational Root Test for a review of this idea).  The rational root test tells us that if the polynomial has a rational root, then it must be contained in this list:

So, we check the easiest one first and plug into the polynomial:

Thus, 1 is not a root and we continue our search with :


Since substituting -1 gives us zero, we have determined that -1 is a root of the polynomial.  Therefore, we may conclude that  or equivalently, , is a factor of .  We know then that  divides  and we carry out that long division here (refer to the tutorial entitled Long Division of Polynomials for a review of this process).


As expected, the remainder is zero and we may write:

Now we focus solely on factoring the quadratic factor that remains on the right hand side.  This can be done either using the techniques explained in the tutorial entitled Solving Quadratic Equations by Factoring or by using the quadratic formula to solve .  In this case, it is not too hard to directly factor this quadratic expression as follows:

Therefore, the complete factorization of can be written:

The nice thing about writing a polynomial in its factored form is that now it is easily seen that its roots are -1, -2 and 3.  We can also conclude that the graph of this polynomial crosses the x-axis three times, once at each of the roots.

In some cases, however, it is not possible to factor a polynomial completely over the real numbers.  For instance, the polynomial  factors partially to , but there is no way to factor the quadratic expression  using real numbers.  Such a polynomial would still have three roots (one real and two complex), but its graph would only cross the x-axis at the point (-1, 0) since x = -1 is the only real zero.