Factor $t^3-6t^2+9t-4$

Question

Factor $t^3-6t^2+9t-4.$

I have studied how to factor only quadratic polynomials. Can you give me a hint?

Answer 1

Whenever you see a polynomial of more than degree $2$ that you have to factor, first use the Rational Root Theorem. It states that, for an integer-coefficient polynomial $a_{n}x^{n} + a_{n-1}x^{n-1} +...+a_{1}x + a_{0}$, all rational rational roots are of the form $\frac{d_{0}}{d_{n}}$, where $d_{0}$ is a factor of $a_{0}$ and $d_{n}$ is a factor of $a_{n}$. (Note: factors can be positive or negative). Then, all possible rational roots of your polynomial are $-4, -2, -1, 1, 2, 4$. We find that $1$ and $4$ are roots, and thus:

$$t^{3}-6t^{2} + 9t - 4 = (t - 1)(t - 4)(t - r)$$

Where $r$ is the third root. Note that $(-1)(-4)(-r)$ must equal $-4$, so $-r = -1$ and thus $r = 1$. Then, the only roots are $\boxed{1\text{ and }4.}$

Answer 2

To my knowledge, there is no 'easy algorithm' to factor cubics or higher order polynomials into factors. The general method is to try to find a root by trial and error, as @Lion Hear already wrote in the comment.