Factoring polynomials (§§5.6&5.7)

Suppose you have a polynomial whose coefficients are all rational numbers. The usual way of factoring such a polynomial is factoring over the integers; it follows the following rules:

The first factor must be a constant, except that (unless it is the only factor) we leave it out if it is 1 or use just a minus sign if it is −1.
Every other factor must be a non-constant polynomial with integer coefficients and a positive leading coefficient.
No factor's coefficients may all be integer multiples of an integer greater than 1.
No factor may be a product of two non-constant polynomials with integer coefficients.

The last of these is the real meat of factoring a polynomial; the other rules are mainly a bookkeeping device to ensure that the factorization is in a standard form.

But these can be done in other ways; factoring over the rational numbers follows these rules:

The first factor must be a constant, except that (unless it is the only factor) we leave it out if it is 1 or use just a minus sign if it is −1.
Every other factor must be a non-constant polynomial with rational coefficients and a leading coefficient of 1.
No factor may be a product of two non-constant polynomials with rational coefficients.

The key difference is that instead of a factor like (3x − 2), we want a factor like (x − ⅔) instead. When factoring over the integers, you should turn x − ⅔ into ⅓(3x − 2); but when factoring over the rational numbers, you should turn 3x − 2 into 3(x − ⅔). The value of this is that you immediately see that ⅔ is a root of the polynomial. Also, the constant coefficient out front will always be the leading coefficient of the polynomial.

If the polynomial might have irrational coefficients, we can be even more general. Given a polynomial with real coefficients, factoring over the real numbers follows these rules:

The first factor must be a constant, except that (unless it is the only factor) we leave it out if it is 1 or use just a minus sign if it is −1.
Every other factor must be a non-constant polynomial with real coefficients and a leading coefficient of 1.
No factor may be a product of two non-constant polynomials with real coefficients.

The difference now is simply that we allow arbitrary real coefficients, which means that we might be able to factor further what we weren't able to factor before. In fact, the non-constant factors will all be either linear or quadratic; this is the Fundamental Theorem of Algebra.

Finally, given a polynomial with complex coefficients, factoring over the complex numbers follows these rules:

The first factor must be a constant, except that (unless it is the only factor) we leave it out if it is 1 or use just a minus sign if it is −1.
Every other factor must be a non-constant polynomial with complex coefficients and a leading coefficient of 1.
No factor may be a product of two non-constant polynomials with complex coefficients.

The difference now is that we allow arbitrary complex coefficients, possibly imaginary ones, which means that we might be able to factor even further. In fact, the non-constant factors will all be linear, again thanks to the Fundamental Theorem of Algebra (since any quadratic factors from the previous step have imaginary roots given by the Quadratic Formula).

For example, consider 3x⁵ − 2x⁴ − 3x³ + 2x² − 6x + 4. Using synthetic division and then a substitution for x², you can factor this over the integers as (3x − 2)(x² − 2)(x² + 1). If you factor out the leading coefficient 3 from the first factor, this is factored over the rational numbers as 3(x − ⅔)(x² − 2)(x² + 1). If you factor x² − 2 as a difference of squares (since 2 is the square of the irrational number √2), this is factored over the real numbers as 3(x − ⅔)(x − √2)(x + √2)(x² + 1). Finally, if you factor x² + 1 as a difference of squares (since −1 is the square of the imaginary number i), this is factored over the complex numbers as 3(x − ⅔)(x − √2)(x + √2)(x − i)(x + i). In summary, we have these five forms of the polynomial, each more fully factored than the ones before:

Expanded:: 3x⁵ − 2x⁴ − 3x³ + 2x² − 6x + 4;
Factored over the integers:: (3x − 2)(x² − 2)(x² + 1);
Factored over the rational numbers:: 3(x − ⅔)(x² − 2)(x² + 1);
Factored over the real numbers:: 3(x − ⅔)(x − √2)(x + √2)(x² + 1);
Factored over the complex numbers:: 3(x − ⅔)(x − √2)(x + √2)(x − i)(x + i).

Notice how this last form has the leading coefficient and the 5 complex roots laid out in a row. This is the form that the textbook will sometimes ask for in the problem sets.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2026 April 20. Toby reserves no legal rights to it.

The permanent URI of this web page is https://tobybartels.name/MATH-1150/2026SP/factoring/.