2.8. Zeroes of Rational Functions http://www.ck12.org
f(x) =^6 x^3 −^7 x−x^21 −x+^2- Identify the zeroes and holes of the following rational function.
f(x) =^2 (x+^12 )((xx++^11 )()x+^1 )
Answers: - There are an infinite number of functions that meet the requirements. An illustrative example would be:
f(x) = (x− 1 )(x+ 1 )·((xx−− 11 )()(xx++^11 ))
The twoxvalues that are holes are identical to the twoxvalues that would be zeroes. Therefore, this function has
no zeroes because holes exist in their place. - After noticing that a possible hole occurs atx=1 and using polynomial long division on the numerator you
should get:
f(x) = ( 6 x^2 −x− 2 )·xx−−^11
A hole occurs atx=1 which turns out to be the point (1, 3) because 6· 12 − 1 − 2 =3.
They-intercept always occurs wherex=0 which turns out to be the point (0, -2) becausef( 0 ) =−2.
To find thex-intercepts you need to factor the remaining part of the function:
( 2 x+ 1 )( 3 x− 2 )
Thus the zeroes (x-intercepts) arex=−^12 ,^23. - The hole occurs atx=−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 thexvalues of either the zeroes or holes of a function.
Practice
Identify the intercepts and holes of each of the following rational functions.
- f(x) =x^3 +x^2 x−−^102 x+^8
2.g(x) =^6 x^3 −^17 x−x^23 −^5 x+^6
3.h(x) =(x+^2 x)(− 11 −x) - j(x) =(x−^4 )(xx++^22 )(x+^2 )
5.k(x) =x(x−^3 )(x−(x^4 −)( 3 x)(+x^4 +)( 4 x)+^4 )(x+^2 ) - f(x) =x(x+(x^1 −)( 1 x)(+x^1 +)( 1 x)−^1 )
7.g(x) =x^3 −xx (^22) −− 1 x+^1
8.h(x) =^4 x−−x 22
- Create a function with holes atx= 3 , 5 ,9 and zeroes atx= 1 ,2.
- Create a function with holes atx=− 1 ,4 and zeroes atx=1.
- Create a function with holes atx= 0 ,5 and zeroes atx= 2 ,3.
- Create a function with holes atx=− 3 ,5 and zeroes atx=4.
- Create a function with holes atx=− 2 ,6 and zeroes atx= 0 ,3.
- Create a function with holes atx= 1 ,5 and zeroes atx= 0 ,6.
- Create a function with holes atx= 2 ,7 and zeroes atx=3.