To determine the total number of roots of each polynomial, we need to examine their degrees. The degree of a polynomial gives us the maximum number of roots, assuming we consider complex roots as well.
### Polynomial 1: [tex]\( f_1(x) = 3x^6 + 2x^5 + x^4 - 2x^3 - 4 \)[/tex]
1. Identify the degree:
The highest power of [tex]\( x \)[/tex] in the polynomial is 6, which is the degree of the polynomial.
2. Number of roots:
A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (including multiplicities and considering complex roots).
Therefore, the polynomial [tex]\( f_1(x) = 3x^6 + 2x^5 + x^4 - 2x^3 - 4 \)[/tex] has 6 roots.
### Polynomial 2: [tex]\( f_2(x) = (3x^4 + 1)^2 \)[/tex]
1. Expand to find the degree:
This polynomial is a square of another polynomial. The inside polynomial is [tex]\( 3x^4 + 1 \)[/tex], which has a degree of 4. When you square it, the degree doubles.
[tex]\[
\left(3x^4 + 1\right)^2 \implies \text{Degree} = 2 \times 4 = 8
\][/tex]
2. Number of roots:
A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots.
Therefore, the polynomial [tex]\( f_2(x) = (3x^4 + 1)^2 \)[/tex] has 8 roots.
### Polynomial 3: [tex]\( g(x) = 5x - 12x^2 + 3 \)[/tex]
1. Identify the degree:
The highest power of [tex]\( x \)[/tex] in the polynomial is 2, which is the degree of the polynomial.
2. Number of roots:
A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots.
Therefore, the polynomial [tex]\( g(x) = 5x - 12x^2 + 3 \)[/tex] has 2 roots.
### Summary
The total number of roots for each polynomial is as follows:
- [tex]\( f_1(x) = 3x^6 + 2x^5 + x^4 - 2x^3 - 4 \)[/tex] has 6 roots.
- [tex]\( f_2(x) = (3x^4 + 1)^2 \)[/tex] has 8 roots.
- [tex]\( g(x) = 5x - 12x^2 + 3 \)[/tex] has 2 roots.
Hence, the result is: [tex]\( (6, 8, 2) \)[/tex].
