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Zeroes of Rational Functions

Values where the numerator equals zero but the denominator doesn't.

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Zeroes of Rational Functions

The zeroes of a function are the collection of x values where the height of the function is zero.  How do you find these values for a rational function and what happens if the zero turns out to be a hole

Finding Zeroes of Rational Functions

Zeroes are also known as x-intercepts, solutions or roots of functions.  They are the x values where the height of the function is zero.  For rational functions, you need to set the numerator of the function equal to zero and solve for the possible x values.  If a hole occurs on the x value, then it is not considered a zero because the function is not truly defined at that point. 

Take the following rational function:


Notice how one of the x+3 factors seems to cancel and indicate a removable discontinuity.  Even though there are two x+3 factors, the only zero occurs at x=1 and the hole occurs at (-3, 0).

Watch the video below and focus on the portion of this video discussing holes and x-intercepts.


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 x values. When a hole and a zero occur at the same point, the hole wins and there is no zero at that point. 

Example 2

Create a function with zeroes at x=1,2,3 and holes at x=0,4

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 y intercepts of the following rational function without graphing. 


The holes occur at x=1,1.  To get the exact points, these values must be substituted into the function with the factors canceled.


The holes are (-1, 0); (1, 6).  The zeroes occur at x=0,2,2.  The zero that is supposed to occur at x=1 has already been demonstrated to be a hole instead. 

Example 4

Identify the y intercepts, holes, and zeroes of the following rational function. 


After noticing that a possible hole occurs at x=1 and using polynomial long division on the numerator you should get:


A hole occurs at x=1 which turns out to be the point (1, 3) because 61212=3

The y-intercept always occurs where x=0 which turns out to be the point (0, -2) because f(0)=2.

To find the x-intercepts you need to factor the remaining part of the function:


Thus the zeroes (x-intercepts) are x=12,23.

Example 5

Identify the zeroes and holes of the following rational function. 


The hole occurs at x=1 which turns out to be a double zero.  The hole still wins so the point (-1, 0) is a hole.  There are no zeroes.  The constant 2 in front of the numerator and the denominator serves to illustrate the fact that constant scalars do not impact the x values of either the zeroes or holes of a function. 


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

  1. f(x)=x3+x210x+8x2
  2. g(x)=6x317x25x+6x3
  3. h(x)=(x+2)(1x)x1
  4. j(x)=(x4)(x+2)(x+2)x+2
  5. k(x)=x(x3)(x4)(x+4)(x+4)(x+2)(x3)(x+4)
  6. f(x)=x(x+1)(x+1)(x1)(x1)(x+1)
  7. g(x)=x3x2x+1x21
  8. h(x)=4x2x2
  9. Create a function with holes at x=3,5,9 and zeroes at x=1,2.
  10. Create a function with holes at x=1,4 and zeroes at x=1
  11. Create a function with holes at x=0,5 and zeroes at x=2,3.
  12. Create a function with holes at \begin{align*}x=-3, 5\end{align*} and zeroes at \begin{align*}x=4\end{align*}
  13. Create a function with holes at \begin{align*}x=-2, 6\end{align*} and zeroes at \begin{align*}x=0, 3\end{align*}.
  14. Create a function with holes at \begin{align*}x= 1, 5\end{align*} and zeroes at \begin{align*}x=0,6\end{align*}.
  15. 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. 

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Hole A hole exists on the graph of a rational function at any input value that causes both the numerator and denominator of the function to be equal to zero.
Rational Expression A rational expression is a fraction with polynomials in the numerator and the denominator.
Rational Function A rational function is any function that can be written as the ratio of two polynomial functions.
Zero The zeros of a function f(x) are the values of x that cause f(x) to be equal to zero.

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