Question

How do you factor completely `t^3+t^2-22t-40`?

Answer 1

`t^3+t^2-22t-40=color(green)((t+2)(t+4)(t-5))`

Explanation:

Using the Rational Root Theorem, the possible roots of the given polynomial are contained in the set:
`color(white)("XXX"){+-1,+-2,+-4,+-8,+-10,+-20}`

Evaluating the given polynomial for each possible root (I chose to use a spread sheet to do this; see below)
we can determine the roots: `-2, -4, and +5`
`color(white)("XXX")`note that since the polynomial is of degree 3
`color(white)("XXXXX")`there can be a maximum of 3 unique roots.

which implies the factors:
`color(white)("XXX")(t+2)(t+4)(t-5)`