Of course, let's determine the number of roots for each polynomial function provided.
1. For the polynomial [tex]\( f(x) = 2x^3 + x^2 - 7x + 1 \)[/tex]:
According to the Fundamental Theorem of Algebra, the number of roots of a polynomial is equal to its degree (the highest power of [tex]\(x\)[/tex] in the polynomial).
In this polynomial, the highest power of [tex]\(x\)[/tex] is 3, which is the degree of the polynomial.
Therefore, [tex]\( f(x) = 2x^3 + x^2 - 7x + 1 \)[/tex] has 3 roots.
2. For the polynomial [tex]\( f(x) = -3x + 5x^2 + 8 \)[/tex]:
Similarly, we look at the highest power of [tex]\(x\)[/tex] in this polynomial.
The highest power of [tex]\(x\)[/tex] is 2, indicating that the degree of the polynomial is 2.
Therefore, [tex]\( f(x) = -3x + 5x^2 + 8 \)[/tex] has 2 roots.
3. For the polynomial [tex]\( f(x) = (x^2 + 6)^2 \)[/tex]:
First, identify the highest power of [tex]\(x\)[/tex] inside the parentheses, which in this case is [tex]\(x^2\)[/tex].
Since this term is squared again as an outer function, we multiply the degrees: [tex]\(2 \times 2 = 4\)[/tex].
Therefore, [tex]\( f(x) = (x^2 + 6)^2 \)[/tex] has 4 roots.
By the Fundamental Theorem of Algebra, we have determined the number of roots for each polynomial function provided:
[tex]\( f(x) = 2x^3 + x^2 - 7x + 1 \)[/tex] has 3 roots.
[tex]\( f(x) = -3x + 5x^2 + 8 \)[/tex] has 2 roots.
[tex]\( f(x) = (x^2 + 6)^2 \)[/tex] has 4 roots.
