Let's find the total number of roots for each given polynomial function.
### For [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex]:
1. Identify the polynomial: The given polynomial is of degree 6. This tells us that it can have up to 6 roots (including multiplicities).
2. Find the roots: Solving [tex]\( 3x^6 + 2x^5 + x^4 - 2x^3 = 0 \)[/tex] reveals that there are indeed 6 roots when considering multiplicities.
So, the total number of roots for [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] is:
[tex]\[ \boxed{6} \][/tex]
### For [tex]\( g(x) = 5x - 12x^2 + 3 \)[/tex]:
1. Identify the polynomial: The given polynomial is of degree 2. This tells us that it can have up to 2 roots.
2. Find the roots: Solving [tex]\( 5x - 12x^2 + 3 = 0 \)[/tex] reveals that there are exactly 2 roots.
So, the total number of roots for [tex]\( g(x) = 5x - 12x^2 + 3 \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
