according to the fundamental theorem of algebra, which polynomial function has exactly 6 roots?\n$f(x)=5x^{4}…

Explanation:

Step1: Recall the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that the number of roots of a polynomial function (f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0) (where (a_n\neq0)) is equal to its degree (n).

Step2: Determine the degree of each polynomial

• For (f(x)=5x^4 + 10x^2+2), the highest - power of (x) is (4), so the degree is (4) and it has (4) roots.
• For (f(x)=5x^5+3x^4 + 12x^3+7x^2-2x + 15), the highest - power of (x) is (5), so the degree is (5) and it has (5) roots.
• For (f(x)=6x^5+x^3-4x^2+x - 5), the highest - power of (x) is (5), so the degree is (5) and it has (5) roots.
• For (f(x)=7x^6+3x^3+12), the highest - power of (x) is (6), so the degree is (6) and it has (6) roots.

Answer:

(f(x)=7x^6 + 3x^3+12)