RATIONAL
FUNCTIONS
x + 2x − 5x − 1
=
x−2
Quotient: + 4 − 3 +
DIVISION ALGORITHM
This is how you write your answers after dividing the functions
quotient
= + or
dividend
divisor
= +
remainder
REMAINDER THEOREM
If a polynomial is divided by − , then the remainder is = .
Example: Evaluate = 3 + 8 + 5 − 7 when = −2.
= − = 3−2
+ 8−2
+ 5−2 − 7 = So, the remainder for this polynomial is 9.
You can double check this method by finding the remainder using synthetic or long division.
List all possible rational zeros of = 2 + 3 − 8 + 3
!=all factors of constant term (in this case, it is 3)
= all factors of the leading coefficient (in this case, it is 2)
Possible zeros:
1 3
! ±1, ±3
=
= ±1, ±3, ± , ± 2 2
±1, ±2
These are possible zeros; you must use synthetic division to find the real zeros. You can start with any possible
zero.
Using x=1:
1
2 3 -8 3
2 5 -3
2
5 -3
0
Once you have a remainder of zero, the x-value is a solution.
Then you take the solution, set equal to 0 and factor it to find the
remaining zeros.
2 + 5 − 3 = 2 − 1 + 3
All factors of = − 12 − 1 + 3 Real zeros of : = 1,
$
, −3
FINDING VERTICAL, HORIZONTAL AND SLANT ASYMPTOTES
Horizontal Asymptote
• If degree of denominator > degree of numerator, f(x) has a horizontal asymptote at y=0. • If degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
• If degree of denominator < degree of numerator, there is no horizontal asymptote. Vertical Asymptote
• Set denominator equal to 0 and solve for x (same method one uses to find the domain).
Slant Asymptote
• Only occurs when degree of numerator is exactly one more degree than the degree of the denominator.
• To find, use long division and only the quotient will be the slant asymptote; NOT THE REMAINDER!
How to Find Holes:
1. Always factor numerator and denominator and reduce the fraction first. If the fraction does not reduce,
there is no hole in its graph.
2. IF THE FRACTION REDUCES, set the factor(s) that cancel out equal to zero and solve for x. This
value is the x-coordinate(s) of one or more holes in the graph. The y-coordinate is calculated by
substituting the x-coordinate into the REDUCED FRACTION.