Factorize the given equation.

Question

Factorize $$f(t) = t^3 - 11t^2 - 39t - 45$$

Assuming the above polynomial has a rational root, I tested the above equation using $+1$, $-1$, $+2$, $-2$. These did not work out. Then I tried $t =3$. Which gave me the factorization $$(t - 3)^2 (t - 5)$$

I want to know if there is a way to figure out a list of rational numbers to be tested for the given equation? Currently, I randomly pick up the first few integers and test to see if any one of them satisfies the given equation.

Answer 1

I believe what you are looking for is the Rational Root Theorem.

It states that for a polynomial $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$ with leading coefficient $a_n$ and constant term $a_0$ that all rational roots, when written as a fraction in simplest form $\frac{p}{q}$ satisfy:

1. $p$ is a factor of $a_0$.

2. $q$ is a factor of $a_n$.

Thus, we can generate a list of possible rational roots for any polynomial. In this case, the possible rational roots are: $\pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45$.

Notice that in this case the leading coefficient is $1$, so all the possible rational roots will in fact be integers.

To use this, you must first make sure that all the polynomial coefficients are integers (if they are rational, you can make them integers by multiplying).

Answer 2

Test $\frac{p}{q}$ where $p|a_0$ and $q|a_n$. $a_0$ is the constant coefficient and $a_n$ is the coefficient of highest order term.