Why might it be useful to know the linear factors of a polynomial function?

It is useful to know the linear factors of a polynomial because they give you the zeros of the polynomial. If (x-c) is one of the linear factors of a polynomial, then p(c)=0.

Here the notation p(x) is used to denoted a polynomial function at p(c) means the value of that function when evaluated at c.

Conversely, if d is a zero of the polynomial, then (x-d) is a factor.