10.2 - Factoring polynomials
You should read section 10.1, Introduction to polynomials,
before reading this section.
Factoring polynomials using the deflation method
Let f (x) denote the polynomial of degree n that we wish to factor.
Also, let f (x) = 0 be the corresponding polynomial equation and let
y = f (x) be the corresponding graph of the polynomial function.
The deflation method starts by either:
Let r denote the value found. Then r is a root
of the polynomial equation and (x − r) is a factor of the polynomial.
This means that the polynomial can be written as
- using trial and error (trying simple values like x = 0, 1, −1, 2, −2, … ) and
finding a value of x that causes the polynomial to vanish
(i.e. causes f (x) to equal zero), or
- inspecting the graph and finding a value of x where the graph crosses
or touches the x axis.
polynomial = (x − r) · (other factor).
Then we find the other factor by dividing the polynomial by (x − r).
This is called deflating the polynomial.
(We saw how to divide polynomials in section 3.5.)
The process is then repeated with the deflated polynomial.
The process stops (fails) when no more zeros of the deflated polynomial can be found by
trial and error and the graph of the deflated polynomial has no more crossings of the x axis.
Note on graph touching the x axis: Suppose that the graph
touches, rather than crosses, the x axis at x = 3. Then x = 3 is a double root and
(x − 3) 2 is a factor.
In order to divide this factor into the polynomial to deflate it, we must write it in the
expanded form x 2 − 6 x + 9.
Example: Factor the polynomial
16 x 3 − 13 x − 3.
Solution: It is not hard to see that x = 1 is a zero of this polynomial.
This means that x − 1 is a factor. Divide this factor into the polynomial:
This means that the other factor (the deflated polynomial) is the quadratic
16 x 2 + 16 x + 3. It can be factored further
by the deflation method but it is easier to use the
method of section 8.3. As a result of that factoring,
we get this final result:
(x − 1) (4x + 1) (4 x + 3).
Factoring polynomials numerically
Once the deflation method fails, the deflated polynomial can theoretically
still be further factored over the real numbers into quadratic factors
or over the complex numbers into linear factors containing complex numbers.
Unfortunately there is no way to do this using algebra. However there
is a numerical method, due to Laguerre, to do it. The Algebra Coach uses his method when the
deflation method fails.