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PRECALCULUS
Polynomial
and
Rational
Functions
Factoring
1. Common Factors: Using the distributive property to
separate common factors.
Ex:
ax  ay  az  a( x  y  z )
2. Difference of squares:
2
2
x

y
 ( x  y )( x  y )
EX:
3. Difference of cubes:
Ex: x3  y3   x  y  ( x 2  xy  y 2 )
4. Sum of cubes:
3
3
2
2
x

y

x

y
(
x

xy

y
)
Ex:


Factoring Continued
5. Perfect Square:
2
2
2
x

2
xy

y

(
x

y
)
Ex:
x 2  2 xy  y 2  ( x  y )2
6. Perfect Cube:
3
2
2
3
3
x

3
x
y

3
xy

y

(
x

y
)
Ex:
x3  3x 2 y  3xy 2  y 3  ( x  y )3
7. Grouping:
Ex: 9 x3  18 x 2  x  2  9 x 2 ( x  1)  1 x  2 
 (9 x 2  1)( x  1)
 (3x  1)(3x  1)( x  1)
8. Binomial Product:
2
2
acx

(
ad

bc
)
xy

bdy
 (ax  by )(cx  dy )
Ex:
Synthetic Division…
NOTE: only works with first degree divisors
1. Solve D(x) =0 for divisor.
2. Place coefficients in descending order with placeholders for
missing terms
3. Place first coefficient on the third line.
2
1
1
3
-12
5
-2
2
10
-4
2
5
-2
1
0
Synthetic Division…
Ex: Divide :
x 4  3 x3  12 x 2  5 x  2
 x3  5 x 2  2 x  1
x2
2 1 3 -12 5 -2
2 10 -4 2
1 5 -2 1 0
Ex:
Divide :
21
x5  3 x 2  1
4
3
2
 x  2 x  4 x  5 x  10 
x2
x2
2
1 0 0 -3 0 1
2 4 8 10 20
1 2 4 5 10 21
Graphing Polynomial Functions
Remainder Theorem: When a polynomial f(x) is divided by x-r,
the remainder is f(r).
Px
R(x)
= Q x +
Dx
Dx
If D(x) = (x-r):
P(x) = Q(x)(x - r) +R
P(r) = Q(r)(r - r) +R
P(r) = Q(r)(0) +R = R
P  x  = Q  x  D  x  +R  x 
Polynomial Theorems
Intermediate Value Theorem: If f is a polynomial
function on [a,b] such that f(a)  f(b), then f(x) takes on
every value between f(a) and f(b) over the interval [a,b].
This theorem allows you to connect the points found by
synthetic division in order to form a smooth curve.
Root Limitation Theorem: A polynomial function P(x) of
degree n has, at most, n distinct zeros.
A root is a solution for an equation. May be real or imaginary.
Zero is a real root for P(x) = 0. An x-intercept
P( x)  x 2  x  2  ( x  2)( x  1)  0
Zeros at 2 and -1
Polynomial Theorems
Factor Theorem: If r is a root of the polynomial equation
P(x), if and only if x-r is a factor of P(x).
Location Theorem: If f(x) is a polynomial function and f(a)
and f(b) are opposite in sign, then there is at least one real zero
between a and b.
Root may be rational or irrational.
Rational Roots Theorem: If P(x) = anxn + ...+a0 has integer
coefficients and p/q is a rational zero, then p is a factor of a0
and q is a factor of an.
(ax  b)(cx  d )  acx 2  (bc  ad )  bd
Possible roots are (b, b/a, b/c, b/ac, d, d/a, d/c, d/ac
1 a, 1/c, and 1/ac)
Test possible roots by synthetic division.
Root Limitation Theorem…
Ex: List all possible rational roots of:
1. x3  x 2  4 x  4  0
p = 1, 2, 4
q = 1
p 1 1 1 1 2 2 2 2 4 4 4 4
: , , , , , , , , , , ,
q 1 1 1 1 1 1 1 1 1 1 1 1
2. 4 x 4  8 x3  43x 2  29 x  60  0
p = 1, 2, 3,  4, 5, 6, 10, 12, 15, 20, 30
q = 1, 2, 4
p/q = (1, 1/2, 1/4, 2, 3, 3/2, 3/4, 4, 5, 5/2, 5/4, 6, 10,
12, 15, 15/2,15/4, 20, 30, 60)
Polynomial Theorems
Upper and Lower Bound Theorem: If a>0 and all numbers
have the same sign then a is an upper bound. If b<0 and the
signs alternate, b is a lower bound
Ex: Solve
2 x 4  5 x3  8 x 2  25 x  10  0
p/q = (1, 1/2, 2, 5, 5/2, 10)
4
3
2
Ex: 4x - 8x - 43x + 29x + 60 = 0
p/q = (1, 1/2, 1/4, 2, 3, 3/2, 3/4, 4, 5, 5/2, 5/4, 6, 10,
12, 15, 15/2,15/4, 20, 30, 60)
4
-8
-43
29
60
-28
97 -456 2340
-5 4
-20
17
-22 126 -3 is a lower bound
-3 4
-2 4
-16
-11
51 -42 root between -2 & -3
-1 4
-12 -31
60
0 -1 is a zero
1 4
-4
-47
-18
2 4
0
-43
-57
4 -31
-64
3 4
4
-15
0
4 4
Solutions: -1, 4, -5/2, 3/2
Factorable quadratic
 2x - 3  2x +5 
Graphing Rational Functions
Asymptote: A line which a graph approaches but never touches.
Types: Vertical
Horizontal: a Limit
Oblique or slant
Vertical asymptote: Located anywhere the x-value makes the
denominator zero
Horizontal Asymptote: for
P( x)
D( x)
If the degree of P(x) is less than D(x) then y=0 is the asymptote.
If the degree of P(x) is equal to D(x), then y equals the ratio of
the lead coefficients is the asymptote.
Slant asymptotes: Degree of P(x) exceeds D(x) by one. Divide and
the quotient without the remainder is the slant
asymptote.
x3  2 x  3
1
3x  8

x

1

2 x2  4 x  1 2
2(2 x 2  4 x  1)
1
The horizontal asymptote is: y  x  1
2
“Hole in the graph”: If there is a common factor in the
numerator and denominator, the function
will not exist at the x which makes the
denominator zero.
1 
Ex: Find: Limit 

x  x  3 
Ans: 0
 2 x 2  3x  5 
Ex: Find: Limit  2

x  x  x  2 
Ans: 2
 x 1 
Limit
Ex: Find:


x1  x 2  1 
Ans: DNE
 x 1 
Limit
Ex: Find: x1  2 
 x 1
Ans: 1/2
Graphing Rational Functions
1. Find asymptotes
2. Find intercepts
3. Does graph cross horizontal asymptote?
4. Check symmetry
5. Graph selected points
Ex: Graph: f ( x) 
2 x2  3x  5
x2  x  2
1. Asymptotes: x = 2, x = -1, and y = 2
2. Intercepts: y-int=-5/2
x-int DNE
3.
2=
2x 2 - 3x + 5
x2 - x - 2
2x 2 - 2x - 4 = 2x 2 - 3x + 5
-2x - 4 = -3x +5
x9
Exponential and Logarithmic Functions
Transcendental Functions: cannot be derived from simple
algebraic functions.
Exponential: function where a finite number is raised
to the power of x.
f(x) = abx
x is the exponent and b is the base
Exponential Property of Equality: If bx = by and b1, x = y.
Rational Exponents:
Rules of exponents:
m
bn
n
1
 
 bm


b0  1
1
n
b  n
b
 
  b

1
n m
n m
 b 
 b
n
m
Graphs of Exponential Functions:
Ex: y = 2x
0
2
1 x


Ex: y =(1/2)










1 10  1
 22  2
 2121  41
2
2
3
2
 1 281
2 1 4
 21 1 12
2
2 2 1
 2  2  4
1
4
2
Ex: f ( x)  10 x
Ex: f ( x)  2 x1
Ex: f ( x)  2 x  1
Ex: f ( x)  2(3 x )  (23 ) x  8 x
Ex: f ( x)  2 x
Ex: f ( x)  2 x
Ex: f ( x)  2
 x2
Logarithmic Functions
y=2x and y=log2 x
Ex: Graph y  log6 x
6y  x
log 6 y  log x
y log 6  log x
y
log x log x

log 6 .7782
Ex: Graph y  log 6 ( x  2)
Ex: Graph y  log 6 ( x)  2
Ex: Graph
y  log6 (2 x)
Ex: Graph
y  log6 (2 x)
Ex: Graph y  2log6 ( x)
Ex: Graph y   log3 (3x  4)  2
y   log 3 (3 x  4)  2
y  2   log 3 (3 x  4)
2  y  log 3 (3 x  4)
32 y  ( 3 x  4)
log 32 y  log( 3 x  4)
log( 3 x  4)
log 3
log( 3 x  4)
y  2
log 3
2 y 
y  log3  3x 
y  log3  3x  4 
 log3  3( x  43 )
y   log3  3x  4 
y   log3  3x  4   2
y  log3  3x 
y  log3  3x  4 
 log3  3( x  43 )
y  log3  3x  4 
 log3  3( x  43 )
y   log3  3x  4 
y   log3  3x  4 
y   log3  3x  4   2