(a) Use synthetic division and the factor theorem to determine if \( [x-(3+4 i)] \) is a factor of \( f(x)=x^{2}-6 x+25 \). \( [x-(3+4 i)] \) (Choose one) actor of \( f(x) \) is

To determine if \( x - (3 + 4i) \) is a factor of \( f(x) = x^2 - 6x + 25 \), we can use synthetic division by substituting \( 3 + 4i \) into \( f(x) \). If \( f(3 + 4i) = 0 \), then \( x - (3 + 4i) \) is a factor. Upon evaluating, we find \( f(3 + 4i) = (3 + 4i)^2 - 6(3 + 4i) + 25 = 0 \), confirming that it is a factor. Since complex roots come in conjugate pairs, if \( 3 + 4i \) is a root, then its conjugate \( 3 - 4i \) must be as well. Therefore, the quadratic \( f(x) \) can also be factored as \( (x - (3 + 4i))(x - (3 - 4i)) \). The original polynomial can also be represented as \( (x - 3)^2 + 16 \), emphasizing the role that the center and radius play in the representation of complex roots.