To determine the total number of roots of a polynomial function, you need to find its degree. The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] that appears in the polynomial.
Let’s analyze each polynomial function step-by-step:
1. First Polynomial Function: [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex]
- The polynomial is: [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex]
- To determine the degree, look for the highest power of [tex]\( x \)[/tex] in the polynomial.
- In this polynomial, the highest power of [tex]\( x \)[/tex] is [tex]\( x^6 \)[/tex].
Therefore, the degree of the polynomial [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] is 6. This means that this polynomial has a total of 6 roots (counting multiplicities).
2. Second Polynomial Function: [tex]\( f(x) = (3x^4 + 1)^2 \)[/tex]
- The polynomial is: [tex]\( f(x) = (3x^4 + 1)^2 \)[/tex]
- First, observe the inner polynomial: [tex]\( 3x^4 + 1 \)[/tex]
- The degree of [tex]\( 3x^4 \)[/tex] is 4.
- The entire polynomial is squared.
- When you square a polynomial of degree 4, the degree of the resulting polynomial will be [tex]\( 4 \times 2 = 8 \)[/tex].
Therefore, the degree of the polynomial [tex]\( f(x) = (3x^4 + 1)^2 \)[/tex] is 8. This means that this polynomial has a total of 8 roots (counting multiplicities).
Thus, for the given polynomial functions:
- [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] has 6 roots.
- [tex]\( f(x) = (3x^4 + 1)^2 \)[/tex] has 8 roots.
