according to the fundamental theorem of algebra what must be the degree of a polynomial function with one x-intercepts that has a multipliciy of two and four imagibary solutions

According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicities).

In your example, the polynomial has:

1. One \( x \)-intercept with a multiplicity of 2. This counts as 2 roots.
2. Four imaginary solutions. Since imaginary roots come in conjugate pairs, if there are 4 imaginary solutions, they account for 4 roots.

To find the total degree of the polynomial, we add the contributions from the real and imaginary roots:

• Real roots: 2 (from the \( x \)-intercept with multiplicity 2)
• Imaginary roots: 4

Now, summing these gives:

\[ 2 + 4 = 6 \]

Thus, the degree of the polynomial function must be \( 6 \).