This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole (removable discontinuity) occurs where both numerator and denominator share a common factor that equals zero. In this problem, g(x) = (x²-4)/(x-2) can be factored as g(x) = (x+2)(x-2)/(x-2), showing the common factor (x-2). Choice B is correct because at x = 2, both the numerator (2²-4 = 0) and denominator (2-2 = 0) equal zero, creating the removable discontinuity. Choice A is incorrect as it confuses the zero of the simplified function with the hole's location. To help students: Use the difference of squares factoring pattern. Remind them that after canceling, the hole remains at the canceled factor's zero.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function u(x) = (x²-25)/(x-5) can be factored as u(x) = (x+5)(x-5)/(x-5), using the difference of squares pattern, revealing a common factor of (x-5). Choice B is correct because when x = 5, both the numerator (25-25=0) and denominator (5-5=0) equal zero, and the (x-5) factor cancels out, creating a hole at x = 5. Choice A is incorrect because x = -5 only makes the numerator zero after cancellation, creating a zero of the simplified function. To help students: Remember that x²-a² = (x+a)(x-a) for difference of squares. Use algebraic cancellation to identify removable discontinuities versus non-removable ones.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function u(x) = (x²-1)/(x+1) can be factored as u(x) = (x+1)(x-1)/(x+1), showing a common factor of (x+1). Choice B is correct because when x = -1, both the numerator (1-1=0) and denominator (-1+1=0) equal zero, and the (x+1) factor cancels, creating a hole at x = -1. Choice A is incorrect because x = 1 only makes the numerator zero after factoring, not the denominator. To help students: Recognize x²-1 as a difference of squares. Pay attention to the sign in the denominator (x+1) to identify the correct hole location.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function g(x) = (x²-4x)/(x-4) can be factored as g(x) = x(x-4)/(x-4), showing a common factor of (x-4). Choice A is correct because when x = 4, both the numerator (16-16=0) and denominator (4-4=0) equal zero, and the (x-4) factor cancels out, creating a hole at x = 4. Choice B is incorrect because x = 0 only makes the numerator zero, resulting in a zero of the function rather than a hole. To help students: Factor the numerator completely before identifying common factors. Use substitution to verify that both numerator and denominator equal zero at the hole location.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function h(x) = (x²-1)/(x-1) can be factored as h(x) = (x+1)(x-1)/(x-1), revealing a common factor of (x-1). Choice B is correct because when x = 1, both the numerator (1-1=0) and denominator (1-1=0) equal zero, and the (x-1) factor cancels out, creating a hole at x = 1. Choice A is incorrect because x = -1 only makes the numerator zero, which creates a zero of the simplified function, not a hole. To help students: Remember the difference of squares factoring pattern (a²-b² = (a+b)(a-b)). Practice identifying and canceling common factors to find removable discontinuities.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function q(x) = (x²-2x-3)/(x-3) can be factored as q(x) = (x-3)(x+1)/(x-3), revealing a common factor of (x-3). Choice B is correct because when x = 3, both the numerator (9-6-3=0) and denominator (3-3=0) equal zero, and the (x-3) factor cancels out, creating a hole at x = 3. Choice D is incorrect because x = -3 doesn't make either the numerator or denominator zero. To help students: Use factoring techniques for trinomials where the leading coefficient is 1. Check your factorization by expanding to verify it matches the original expression.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function v(x) = (x²+3x)/x can be factored as v(x) = x(x+3)/x, showing a common factor of x. Choice C is correct because when x = 0, both the numerator (0+0=0) and denominator (0) equal zero, and the x factor cancels out, creating a hole at x = 0. Choice A is incorrect because x = -3 only makes the numerator zero after cancellation, not creating a hole. To help students: Factor out common terms from the numerator before identifying cancelable factors. Be careful with x = 0 cases, as they often create holes when x appears in both numerator and denominator.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, h(x) = (x²+5x+6)/(x+2) requires factoring the numerator: x²+5x+6 = (x+2)(x+3), revealing the common factor (x+2). Choice A is correct because when x = -2, both the numerator (4-10+6 = 0) and denominator (-2+2 = 0) equal zero, confirming the hole's location. Choice B is incorrect as it identifies where the simplified function h(x) = x+3 equals zero. To help students: Practice factoring quadratic expressions. Emphasize checking that both original numerator and denominator equal zero at the proposed hole location.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function p(x) = (x²+x)/x can be factored as p(x) = x(x+1)/x, showing a common factor of x. Choice C is correct because when x = 0, both the numerator (0+0=0) and denominator (0) equal zero, and the x factor cancels, creating a hole at x = 0. Choice A is incorrect because x = 1 doesn't make the denominator zero. To help students: Factor out common terms from the numerator before simplifying. Remind students that division by zero creates discontinuities, but when both parts are zero, it's removable.

This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function q(x) = (x²-6x+9)/(x-3) has a numerator that factors as (x-3)², so q(x) = (x-3)²/(x-3), with a common factor of (x-3). Choice B is correct because when x = 3, both the numerator (9-18+9=0) and denominator (3-3=0) equal zero, and one (x-3) factor cancels, creating a hole at x = 3. Choice A is incorrect because x = -3 doesn't make either part zero. To help students: Recognize perfect square trinomials like x²-6x+9 = (x-3)². Practice factoring quadratics to identify common factors with the denominator.