The rational function \( r(x)=\frac{(x+5)(x+3)(x-1)}{(x-5)(x+5)(x-6)} \) has - Zeros (x-intercepts) at \( x=\square \) - Vertical asymptotes at \( x=\square \) - A hole at \( x=\square \) - A horizontal asymptote at \( y=\square \)

To analyze the rational function \( r(x)=\frac{(x+5)(x+3)(x-1)}{(x-5)(x+5)(x-6)} \), we will determine the zeros, vertical asymptotes, holes, and horizontal asymptote step by step. ### Step 1: Finding Zeros (x-intercepts) The zeros of the function occur where the numerator is equal to zero. The numerator is \( (x+5)(x+3)(x-1) \). Setting this equal to zero: \[ (x+5)(x+3)(x-1) = 0 \] This gives us the following equations: 1. \( x + 5 = 0 \) → \( x = -5 \) 2. \( x + 3 = 0 \) → \( x = -3 \) 3. \( x - 1 = 0 \) → \( x = 1 \) Thus, the zeros (x-intercepts) are at \( x = -5, -3, 1 \). ### Step 2: Finding Vertical Asymptotes Vertical asymptotes occur where the denominator is equal to zero, provided that these points do not also make the numerator zero (which would indicate a hole). The denominator is \( (x-5)(x+5)(x-6) \). Setting this equal to zero: \[ (x-5)(x+5)(x-6) = 0 \] This gives us the following equations: 1. \( x - 5 = 0 \) → \( x = 5 \) 2. \( x + 5 = 0 \) → \( x = -5 \) 3. \( x - 6 = 0 \) → \( x = 6 \) Since \( x = -5 \) is also a zero of the numerator, it indicates a hole rather than a vertical asymptote. Therefore, the vertical asymptotes are at \( x = 5 \) and \( x = 6 \). ### Step 3: Finding Holes A hole occurs at points where both the numerator and denominator are zero. From our previous calculations, we found that \( x = -5 \) is a zero of both the numerator and the denominator. Thus, there is a hole at \( x = -5 \). ### Step 4: Finding Horizontal Asymptote To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and denominator. - The degree of the numerator \( (x+5)(x+3)(x-1) \) is 3 (since it is a product of three linear factors). - The degree of the denominator \( (x-5)(x+5)(x-6) \) is also 3 (since it is a product of three linear factors). When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of both the numerator and denominator is 1. Thus, the horizontal asymptote is: \[ y = \frac{1}{1} = 1 \] ### Summary of Results - Zeros (x-intercepts) at \( x = -5, -3, 1 \) - Vertical asymptotes at \( x = 5, 6 \) - A hole at \( x = -5 \) - A horizontal asymptote at \( y = 1 \) Final answers: - Zeros: \( x = -5, -3, 1 \) - Vertical asymptotes: \( x = 5, 6 \) - Hole: \( x = -5 \) - Horizontal asymptote: \( y = 1 \)