Today's soundtrack is Crytopsy: Whisper Supremacy, the first technical death metal album that I ever bought. I remember that when I first listened to it, I was not sure what to make of its chaotic assault on my ears, especially since there was a contrast between the hardcore-style vocals and the tech-death instrumentation. I couldn't anticipate the time changes; the riffs were all over the place, and there wasn't a breakdown to be found anywhere. It was nothing like anything I'd listened to up til that point: it is an unstoppable force of frenetic energy. To this day, I'm blown away by Canadian drummer Flo Mournier's performance on this album.
This afternoon, I learned about the Factor Theorem, a specialized branch of the Remainder Theorem (Part 1, Part 2).
The Factor Theorem states that if x-b is substituted into the polynomial P(x) and the remainder = 0, then x-b is a factor of P(x).
An extension of this theorem is the Factor Property, which tells us that if a polynomial has any factor in the form x-b, the number b is a factor of the constant term of the polynomial; thus, b is a root of the polynomial.
However, there is one important caveat to consider here: the Rational Zero Theorem, which says that when the coefficient of the highest-term variable in the polynomial ≠ 1, we must divide each factor of our constant by each factor of the highest term's coefficient to find all potential roots of our equation.
Using the Factor Theorem, the Factor Property, and the Rational Zero Theorem, we can do quite a lot with a polynomial equation, including: finding all potential roots for a polynomial, determining which of those roots are the factors of a polynomial, and determining whether a given value is a root, a factor, or a value unrelated to a polynomial. Let's go through the steps:
How to find all potential roots of a polynomial (using the Factor Property and the Rational Zero Theorem)
Find all factors of the constant
Given the equation P(x) = 4x² - 8x + 3, we can see that the constant is 3.
The factors of 3 are ±1, ±3.
Find all factors of the coefficient of the highest-term variable in the polynomial
In P(x) = 4x² - 8x + 3, 4x² is the highest-term item
The factors of 4 are ±1, ±2, ±4.
Write out, in rational form, each factor of the constant over each factor of the coefficient to find all potential rational roots
Our numerator factors will be ±1 and ±3; our denominator factors will be ±1, ±2, and ±4.
Our potential rational roots are ±1/1, ±1/2, ±1/4, ±3/1, ±3/2, and ±3/4. We can simplify these as ±1, ±1/2, ±1/4, ±3, ±3/2, and ±3/4.
How to use synthetic division to determine whether a potential root is a factor of a polynomial (more efficient in most cases)
Once we know our potential roots (see above), we can determine whether or not they are factors of our equation by using synthetic division.
Set up a synthetic division "box"
Divide synthetically (need help? Read this post for a detailed step-by-step)
Remainder? If yes, the potential root is not a factor. If no, the potential root is a factor.
How to use substitution to determine whether a potential root is a factor of a polynomial (useful in tricky situations, such as when dividing a polynomial by a binomial that includes a square root)
Substitute the potential root in for every instance of x, then determine whether the equation equals 0. If it does, the potential root is a factor; if not, the potential root is not a factor.