Use the factor theorem to determine if the given binomial is a factor of \( f(x) \). \( f(x)=x^{4}-6 x^{3}+6 x^{2}+7 x-8 \) (a) \( x-1 \) (b) \( x+5 \)

To determine if a binomial is a factor of a polynomial using the factor theorem, we need to evaluate the polynomial at the root of the binomial. If \( f(c) = 0 \), then \( x - c \) is a factor. Let's start with \( (a) \) \( x-1 \): 1. The root is \( c = 1 \). 2. Evaluate \( f(1) = (1)^4 - 6(1)^3 + 6(1)^2 + 7(1) - 8 = 1 - 6 + 6 + 7 - 8 = 0 \). Since \( f(1) = 0 \), \( x - 1 \) is a factor of \( f(x) \). Now for \( (b) \) \( x+5 \): 1. The root is \( c = -5 \). 2. Evaluate \( f(-5) = (-5)^4 - 6(-5)^3 + 6(-5)^2 + 7(-5) - 8 = 625 + 750 + 150 - 35 - 8 = 1482 \). Since \( f(-5) \neq 0 \), \( x + 5 \) is not a factor of \( f(x) \).