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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

Watch This

Focus on the portion of this video discussing holes and x -intercepts.

http://www.youtube.com/watch?v=UnVZs2EaEjI James Sousa: Find the Intercepts, Asymptotes, and Hole of a Rational Function


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. 

Example A

Identify the zeroes and holes of the following rational function. 


Solution:  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).

Example B

Identify the zeroes, holes and  y intercepts of the following rational function without graphing. 


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

f(x) &= x(x-2)(x+1)(x+2)\\f(-1) &= 0, f(1)=-6

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 C

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

Solution:  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:


Concept Problem Revisited

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. 


A zero is where a function crosses the x -axis.  It is also known as a root, solution or x -intercept. 

A rational function is a function with at least one rational expression.

A rational expression is a ratio of two polynomial expressions.

Guided Practice

1.  Create a function with holes instead of zeroes. 

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


3. Identify the zeroes and holes of the following rational function. 



1.  There are an infinite number of functions that meet the requirements.  An illustrative example would be:

f(x)=(x-1)(x+1) \cdot \frac{(x-1)(x+1)}{(x-1)(x+1)}

The two  x values that are holes are identical to the two  x values that would be zeroes.  Therefore, this function has no zeroes because holes exist in their place. 

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

f(x)=(6x^2-x-2) \cdot \frac{x-1}{x-1}

A hole occurs at x=1  which turns out to be the point (1, 3) because 6 \cdot 1^2-1-2=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=-\frac{1}{2}, \frac{2}{3} .

3. 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)=\frac{x^3+x^2-10x+8}{x-2}
  2. g(x)=\frac{6x^3-17x^2-5x+6}{x-3}
  3. h(x)=\frac{(x+2)(1-x)}{x-1}
  4. j(x)=\frac{(x-4)(x+2)(x+2)}{x+2}
  5. k(x)=\frac{x(x-3)(x-4)(x+4)(x+4)(x+2)}{(x-3)(x+4)}
  6. f(x)=\frac{x(x+1)(x+1)(x-1)}{(x-1)(x+1)}
  7. g(x)=\frac{x^3-x^2-x+1}{x^2-1}
  8. h(x)=\frac{4-x^2}{x-2}
  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 x=-3, 5  and zeroes at x=4
  13. Create a function with holes at x=-2, 6  and zeroes at x=0, 3 .
  14. Create a function with holes at x= 1, 5  and zeroes at x=0,6 .
  15. Create a function with holes at x=2, 7  and zeroes at x=3 .




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

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.
Rational Function

Rational Function

A rational function is any function that can be written as the ratio of two polynomial functions.


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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