The Factor Theorem

As the Remainder Theorem points out, if you divide a polynomial p(x) by a factor x − a of that polynomial, then you will get a zero remainder.

This is similar to what you learned back in grammar school; namely, if you divide a number by a factor of that number, then you'll get a zero remainder. For example, if you take 15 and divide it by 3, you'll get a zero remainder; the 3 divided into the 15 "evenly" because 3 is a factor of 15. On the other hand, if you divided 15 by 4, you'd get a remainder of 3, because 4 is not a factor of 15.

This is easy enough to understand when we're dealing with simple whole numbers. It can get a bit confusing when we're dealing with polynomial expressions. So let's look again at the Division Algorithm expression for polynomials:

What is the Division Algorithm for Polynomials?

If x − a is indeed a factor of p(x), then the remainder after division by x − a will be zero. That is, r(x) = 0, and you get:

In terms of the Remainder Theorem, this means that, if x − a is a factor of p(x), then the remainder r(x), when we do synthetic division by x = a, will be zero.

What does the Factor Theorem say?

The point of the Factor Theorem is the reverse of the Remainder Theorem: If you synthetic-divide a polynomial by x = a and get a zero remainder, then not only is x = a a zero of the polynomial (courtesy of the Remainder Theorem), but also x − a is a factor of the polynomial. (Why? It's courtesy of the Factor Theorem.)

How do you use the Factor Theorem (in math)?

Just as with the Remainder Theorem, the point of the Factor Theorem is not to do the long division of a given polynomial by a given factor. When faced with a Factor Theorem exercise, you will instead apply synthetic division and then check for a zero remainder. This Theorem isn't repeating what you already know, but is instead trying to make your life simpler.

---

• Use the Factor Theorem to determine whether x − 1 is a factor of f(x) = 2x⁴ + 3x² − 5x + 7.

---

What is an example of factoring with the Factor Theorem?

• Using the Factor Theorem, verify that x + 4 is a factor of f(x) = 5x⁴ + 16x³ − 15x² + 8x + 16.

---

In practice, the Factor Theorem is used when factoring polynomials "completely". Rather than trying various factors by using long division, you will use synthetic division and the Factor Theorem. Any time you divide by a number (that number being a potential root of the polynomial) and get a zero remainder in the synthetic division, this means that the number is indeed a root, and thus "x minus the number" is a factor. Then you will continue the division with the resulting smaller polynomial, repeating the process until you arrive at a linear factor (at which point, you've found all the factors) or a quadratic (to which you can apply the Quadratic Formula).

• Using the fact that −2 and are zeroes of f(x) = 3x⁴ + 5x³ + x² + 5x − 2, factor the polynomial completely.

(If you haven't yet studied complex numbers, then you'd stop at .)