The **zeroes** of a function are the collection of

### Finding Zeroes of Rational Functions

Zeroes are also known as

Take the following rational function:

Notice how one of the

Watch the video below and focus on the portion of this video discussing holes and

### Examples

#### Example 1

Earlier, you were asked how to find the zeroes of a rational function and what happens if the zero is a hole. To find the zeroes of a rational function, set the numerator equal to zero and solve for the

#### Example 2

Create a function with zeroes at

There are an infinite number of possible functions that fit this description because the function can be multiplied by any constant. One possible function could be:

Note that 0 and 4 are holes because they cancel out.

#### Example 3

Identify the zeroes, holes and

The holes occur at

The holes are (-1, 0); (1, 6). The zeroes occur at

#### Example 4

Identify the

After noticing that a possible hole occurs at

A hole occurs at

The

To find the

Thus the zeroes (

#### Example 5

Identify the zeroes and holes of the following rational function.

The hole occurs at

### Review

Identify the intercepts and holes of each of the following rational functions.

f(x)=x3+x2−10x+8x−2 g(x)=6x3−17x2−5x+6x−3 h(x)=(x+2)(1−x)x−1 j(x)=(x−4)(x+2)(x+2)x+2 k(x)=x(x−3)(x−4)(x+4)(x+4)(x+2)(x−3)(x+4) f(x)=x(x+1)(x+1)(x−1)(x−1)(x+1) g(x)=x3−x2−x+1x2−1 h(x)=4−x2x−2 - Create a function with holes at
x=3,5,9 and zeroes atx=1,2 . - Create a function with holes at
x=−1,4 and zeroes atx=1 . - Create a function with holes at
x=0,5 and zeroes atx=2,3 . - Create a function with holes at \begin{align*}x=-3, 5\end{align*} and zeroes at \begin{align*}x=4\end{align*}.
- Create a function with holes at \begin{align*}x=-2, 6\end{align*} and zeroes at \begin{align*}x=0, 3\end{align*}.
- Create a function with holes at \begin{align*}x= 1, 5\end{align*} and zeroes at \begin{align*}x=0,6\end{align*}.
- Create a function with holes at \begin{align*}x=2, 7\end{align*} and zeroes at \begin{align*}x=3\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.8.