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:

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.

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

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 x² − 6 x + 9.

Example: Factor the polynomial 16 x³ − 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 x² + 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).

Algebra Coach Exercises

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.