Question

1. $f(x)=x^8 - 18x^4 + 81$

a) 8
b) 3 or 1
c) 3
d) 6
use the fundamental theorem of algebra to determine the total number of solutions.

Explanation:

Step1: Recall Fundamental Theorem of Algebra

The theorem states that a polynomial of degree $n$ has exactly $n$ complex solutions (counting multiplicities).

Step2: Identify polynomial degree

For $f(x)=x^8 - 18x^4 + 81$, the highest power of $x$ is 8, so degree $n=8$.

Step3: Apply theorem to find solutions

By the Fundamental Theorem of Algebra, the total number of solutions equals the degree of the polynomial.

Answer:

A) 8