Answer:
f(x) = (x² - 7x + 12)/(x³ - 8x² + 20x - 16) = (x - 3)(x - 4)/(x - 2)²(x - 4)
Step-by-step explanation:
∵ The hole at x = 4
∴ The numerator and denominator both have (x - 4)
∵ Horizontal asymptote at y = 0
∴ The degree of the numerator less than the degree of the denominator
∵ The x-intercept = 3
∴ The numerator has (x -3)
∴ The numerator is (x - 3)(x - 4)
∵ The vertical asymptotes at x = 2
∴ The denominator has (x - 2)²
∴ The denominator is (x - 2)²(x - 4)
[We repeat (x - 2) to make the degree of numerator less than the degree of the denominator]
∴ The numerator is (x² - 7x + 12)
∴ The denominator is (x³ - 8x² + 20x - 16)
∴ f(x) = (x² - 7x + 12)/(x³ - 8x² + 20x - 16) = (x - 3)(x - 4)/(x - 2)²(x - 4)
