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