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Advanced Algebra Final Formulas and Facts
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nth Term of an Arithmetic Sequence for all
positive integers n:
Sum of the First n Terms of an Arithmetic
Series for all positive integers n:
tn  t1  (n  1)d
t t 
Sn  n  1 n 
 2 
nth Term of a Geometric Sequence
Sum of the First n Terms of a Geometric
Series
 1 rn 
where r  1
Sn  t1 

1

r


tn  t1r
n 1
where n > 1.
logb mn  logb m  logb n
If m, n, and b are positive real numbers and
b  1.
log b
Factoring Sum of Two Cubes
a 3  b3   a  b  a 2  ab  b 2


Factoring Difference of Two Cubes
a 3  b3   a  b  a 2  ab  b 2


log b m  p log b m
p
m
 log b m  log b n
n
If m, n, and b are positive real numbers
and b  1.
If logb x  logb y , then x = y
log b b x  x and blogb x  x
for x > 0 If b > 0 and b  1
Sum of an Infinite Geometric Series
t
where r  1
S 1
1 r
If m and b are positive real numbers,
and p is a real number
log b x 
log a x
log a b
For any positive real numbers a
x>0
Compound Interest
 r
A(t )  P 1  
 n
nt
Continuously Compounding Formula:
Simple
Growth/Decay:
A  Pe rt
A  P(1  r )t
Rational Root Theorem:
Let P be a polynomial function with integer
coefficients in standard form. If
p is a root
q
of P ( x )  0 , then p is a factor of the
constant term of P and q is a factor of the
leading coefficient of P.
Vertical Asymptotes: If (x – a) is a factor
of the denominator of a rational function,
but not a factor of the numerator, then x = a
is a vertical asymptote of the graph of the
function.
Standard Equation of a Parabola
y  k  a  x  h
2
Complex Conjugate Root Theorem:
If P is a polynomial function with realnumber coefficients and a + bi is a root of
P ( x )  0 , then a – bi is also a root
of P ( x )  0 .
 1 , b  1,
Horizontal Asymptotes: Let R( x) 
P be a
Q
rational function. P and Q are polynomials.
- If the degree of P is less than the degree of
Q, then y = 0 is the equation of the horizontal
asymptote of the graph of R.
- If the degree of P is equal to the degree of
Q, and a and b are the leading coefficients of
P and Q, respectively, then y  a is the
b
Holes: If (x – b) is a factor of the numerator
and denominator of a rational function, then
there is a hole in the graph at x = b, unless
there is a vertical asymptote at x = b. To
find the y-value, substitute the x-value into
the simplified version of the function.
Standard form of the Equation of a Circle
or x  k  a  y  h 2
b  1,
( x  h)   y  k   r
2
2
equation of the horizontal asymptote of the
graph of R.
- If the degree of P is more than the degree of
Q, then graph of R has no horizontal
asymptote.
Standard Equation of an Ellipse
 x  h
2
a
2
2

y k
b
2
2
1
Standard Equation of a Hyperbola
2
2
2
2
 x  h    y  k   1 or  y  k    x  h   1
a2
b2
b2
a2
Permutations and Combinations
n!
n!
n Pr 
n Cr 
r ! n  r  !
 n  r !
Probability of A or B:
Mutually exclusive: P( A or B)  P( A)  P ( B )
Inclusive:
Probability of the Complement of A:
Probability of Independent Events:
Conditional Probability
P( Aand B)
P( B | A) 
P( A)
P( A and B)  P( A) P( B)
P( Ac )  1  P( A)
y
f(x)=bx
x
y =0
x
-1
0
1
y
1/b
1
b
P( A or B)  P( A)  P( B)  P( Aand B)
y
f(x)=logbx
x =0
x
x
1/b
1
b
y
-1
0
1