http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Example B
Identify the zeroes, holes andyintercepts of the following rational function without graphing.
f(x) =x(x−^2 )(x−(x^1 −)( 1 x)(+x^1 +)( 1 x)+^1 )(x+^2 )
Solution: The holes occur atx=− 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 ) =− 6The holes are (-1, 0); (1, 6). The zeroes occur atx= 0 , 2 ,−2. The zero that is supposed to occur atx=−1 has
already been demonstrated to be a hole instead.
Example C
Create a function with holes atx= 1 , 2 ,3 and zeroes atx= 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:
f(x) =(x−^1 )(x−x^2 ()(x−x 4 −)^3 )x(x−^4 )
Concept Problem Revisited
To find the zeroes of a rational function, set the numerator equal to zero and solve for thexvalues. When a hole and
a zero occur at the same point, the hole wins and there is no zero at that point.
Vocabulary
Azerois where a function crosses thex-axis. It is also known as a root, solution orx-intercept.
Arational functionis a function with at least one rational expression.
Arational expressionis a ratio of two polynomial expressions.
Guided Practice
- Create a function with holes instead of zeroes.
- Identify theyintercepts, holes, and zeroes of the following rational function.