1. Factor numerator and denominator. 2. Identify common factors. 3. Set common factor = 0 to find x-coordinate. 4. Cancel common factors. 5. Evaluate simplified function at x-coordinate to find y-coordinate.

1. Identify the x-coordinate of the hole. 2. Cancel the common factor. 3. Evaluate the simplified function at the x-coordinate.

1. Check if (x-a) is a factor of both numerator and denominator. 2. If yes, then a hole exists at x=a.

1. Factor the numerator and the denominator. 2. Identify and cancel the common factors. 3. Write the simplified function.

1. Graph the rational function. 2. Zoom in around the x-coordinate of the hole. 3. Observe the y-value the function approaches.

1. Factor completely. 2. Identify all common factors. 3. Find coordinates for each hole separately.

Factor both the numerator and the denominator completely.

Set the common factor (that appears in both numerator and denominator) equal to zero and solve for x.

Substitute the x-value of the hole into the simplified rational function.

Verify that the function is undefined at that x-value due to a common factor, but the limit exists as x approaches that value.

Given r(x) = p(x)/q(x) with a hole at x=a, y = lim x->a [r(x) after canceling common factors].

If r(x) has a hole at x = c, then $\lim_{x \to c} r(x) = L$, where L is the y-coordinate of the hole.

$x^2 - a^2 = (x - a)(x + a)$

$\frac{(x-a)p(x)}{(x-a)q(x)}$, where p(a) and q(a) are not simultaneously zero.

L = $\lim_{x \to c}$ [Simplified r(x) after canceling common factors]

$(x - a)$, where 'a' is a root of the polynomial.

If the simplified function is s(x), then y-coordinate = s(a), where x=a is the x-coordinate of the hole.

$x^3 - a^3 = (x - a)(x^2 + ax + a^2)$

$x^3 + a^3 = (x + a)(x^2 - ax + a^2)$

$\lim_{x \to a} f(x)$ exists, but $f(a)$ is undefined.

A point where the function is undefined because a factor is present in both the numerator and denominator.

The number of times a zero appears as a factor in the factorization of a polynomial.

A point on a function's graph that can be 'removed' by redefining the function at that point (e.g., a hole).

A function that can be defined as a quotient of two polynomials.

The y-coordinate of the hole is the limit of the function as x approaches the x-coordinate of the hole.

The value that a function approaches as the input (or argument) approaches some value.

The function does not have a defined value at that specific input value, often due to division by zero.

An expression that divides evenly into the polynomial, leaving no remainder.

To divide both the numerator and denominator of a fraction by the same factor, simplifying the expression.

The value 'c' where the common factor (x-c) equals zero.