3 roots, since the polynomial is of third degree.
This follows immediately from the zero product property: if [tex]ab=0[/tex], then either [tex]a=0[/tex] or [tex]b=0[/tex]. We have
[tex](9x + 7)(4x + 1)(3x + 4) = 0[/tex]
from which it follows that
[tex]\begin{cases}9x+7=0\\4x+1=0\\3x+4=0\end{cases}[/tex]
each of which admits only one solution.
Or, using the fundamental theorem of algebra, expanding we have a polynomial that is of third degree:
[tex](9x + 7)(4x + 1)(3x + 4)=108 x^3 + 255 x^2 + 169 x + 28[/tex]
The theorem states that a polynomial [tex]a_nx^n+\cdots+a_1x+a_0[/tex] will have up to [tex]n[/tex] distinct roots. In this case, it follows that there are exactly 3, since the solutions to the system above are all distinct.
