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ALGEBRA
Factors and Zeros of Polynomials
Let px an x n an1x n1 . . . a1x a0 be a polynomial. If pa 0, then a is a zero of the
polynomial and a solution of the equation px 0. Furthermore, x a is a factor of the polynomial.
Fundamental Theorem of Algebra
An nth degree polynomial has n (not necessarily distinct) zeros. Although all of these
zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.
Quadratic Formula
If px ax 2 bx c, and 0 ≤ b2 4ac, then the real zeros of p are x b ± b2 4ac2a.
Special Factors
x 2 a 2 x ax a
x 3 a 3 x ax 2 ax a 2
x 3 a3 x ax 2 ax a 2
x 4 a 4 x 2 a 2x 2 a 2
Binomial Theorem
x y2 x 2 2xy y 2
x y2 x 2 2xy y 2
x y3 x 3 3x 2y 3xy 2 y 3
x y3 x 3 3x 2y 3xy 2 y 3
x y4 x 4 4x 3y 6x 2y 2 4xy3 y 4
x y4 x 4 4x 3y 6x 2y 2 4xy 3 y 4
nn 1 n2 2 . . .
x
y nxy n1 y n
2!
nn 1 n2 2 . . .
x yn x n nx n1y x
y ± nxy n1 y n
2!
x yn x n nx n1y Rational Zero Theorem
If px an x n a n1x n1 . . . a1x a0 has integer coefficients, then every
rational zero of p is of the form x rs, where r is a factor of a0 and s is a factor of an.
Factoring by Grouping
acx 3 adx 2 bcx bd ax 2cx d bcx d ax 2 bcx d
Arithmetic Operations
ab ac ab c
ab a d ad
c
d b c bc
b
ab
a c
c
a
c
ad bc
b d
bd
a
b
a
c
bc
ab a b
c
c
c
ab ba
cd
dc
ab ac
bc
a
a
ac
b
b
c
Exponents and Radicals
a0 1,
ab
x
a0
ax
bx
ab x a xb x
a xa y a xy
n am amn
ax © Houghton Mifflin Company, Inc.
1
ax
a a12
ax
a xy
ay
n a a1n
n
n a
n b
ab axy a xy
ab n
n a
n b
DERIVATIVES AND INTEGRALS
Basic Differentiation Rules
1.
4.
7.
10.
13.
16.
19.
22.
25.
28.
31.
34.
d
cu cu
dx
d u
vu uv
dx v
v2
d
x 1
dx
d u
e eu u
dx
d
sin u cos uu
dx
d
cot u csc2 uu
dx
d
u
arcsin u dx
1 u2
d
u
arccot u dx
1 u2
d
sinh u cosh uu
dx
d
coth u csch2 uu
dx
d
u
sinh1 u dx
u2 1
d
u
coth1 u dx
1 u2
2.
5.
8.
11.
14.
17.
20.
23.
26.
29.
32.
35.
d
u ± v u ± v
dx
d
c 0
dx
d
u
u
u , u 0
dx
u
d
u
loga u dx
ln au
d
cos u sin uu
dx
d
sec u sec u tan uu
dx
d
u
arccos u dx
1 u2
d
u
arcsec u dx
u u2 1
d
cosh u sinh uu
dx
d
sech u sech u tanh uu
dx
d
u
cosh1 u dx
u2 1
d
u
sech1 u dx
u1 u2
3.
5.
7.
9.
11.
13.
15.
17.
kf u du k f u du
2.
du u C
4.
eu du eu C
6.
cos u du sin u C
8.
cot u du ln sin u C
10.
csc u du ln csc u cot u C
12.
csc2 u du cot u C
14.
csc u cot u du csc u C
16.
du
1
u
arctan C
2
u
a
a
18.
a2
© Houghton Mifflin Company, Inc.
6.
9.
Basic Integration Formulas
1.
3.
12.
15.
18.
21.
24.
27.
30.
33.
36.
d
uv uv vu
dx
d n
u nu n1u
dx
d
u
ln u dx
u
d u
a ln aau u
dx
d
tan u sec2 uu
dx
d
csc u csc u cot uu
dx
d
u
arctan u dx
1 u2
d
u
arccsc u dx
u u2 1
d
tanh u sech2 uu
dx
d
csch u csch u coth uu
dx
d
u
tanh1 u dx
1 u2
u
d
csch1 u dx
u 1 u2
f u ± gu du au du ln1aa
u
f u du ±
C
sin u du cos u C
tan u du ln cos u C
sec u du ln sec u tan u C
sec2 u du tan u C
sec u tan u du sec u C
du
u
C
a
du
1
u
arcsec
C
2
2
a
a
uu a
a2 u2
arcsin
gu du
FORMULAS FROM GEOMETRY
Triangle
Sector of Circular Ring
h a sin 1
Area bh
2
(Law of Cosines)
c
p average radius,
w width of ring,
in radians
Area pw
a
θ
h
b
c2 a 2 b2 2ab cos c
(Pythagorean Theorem)
a2
Area ab
a
b2
Equilateral Triangle
Area s
h
4
A
Right Circular Cone
r 2h
3
Lateral Surface Area rr2 h2
h
h
Volume b
h
Area a b
2
h
s
Parallelogram
Trapezoid
a
A area of base
Ah
Volume 3
s
3s2
Area bh
a 2 b2
2
Cone
3s
2
r
Frustum of Right Circular Cone
a
h
a
h
r
rR h
3
Lateral Surface Area sR r
Volume b
b
r2
R2
s
h
Right Circular Cylinder
Circle
r2
Area Circumference 2 r
Volume r 2h
Lateral Surface Area 2 rh
r
Sector of Circle
in radians
r2
Area 2
s r
w
b
Circumference 2
b
h
θ
Ellipse
Right Triangle
c2
p
R
r
h
Sphere
4
Volume r 3
3
Surface Area 4 r 2
s
θ
r
Circular Ring
p average radius,
w width of ring
Area R 2 r 2
2 pw
© Houghton Mifflin Company, Inc.
r
Wedge
r
p
R
w
A area of upper face,
B area of base
A B sec A
θ
B
TRIGONOMETRY
Definition of the Six Trigonometric Functions
Opposite
Right triangle definitions, where 0 < < 2.
opp
hyp
sin csc e
s
u
hyp
opp
en
pot
Hy
adj
hyp
cos sec θ
hyp
adj
Adjacent
opp
adj
tan cot adj
opp
Circular function definitions, where is any angle.
y
r
y
sin csc r = x2 + y2
r
y
(x, y)
x
r
r
cos sec θ
y
r
x
x
y
x
x
cot tan x
y
Reciprocal Identities
1
sin x csc x
1
csc x sin x
1
sec x cos x
1
cos x sec x
sin x
cos x
cot x cos x
sin x
Pythagorean Identities
sin2 x cos2 x 1
1 tan2 x sec2 x
(− 12 , 23 ) π (0, 1) ( 12 , 23 )
90°
(− 22 , 22 ) 3π 23π 2 π3 π ( 22 , 22 )
120°
60°
4 π
45°
(− 23 , 12) 56π 4150°135°
( 23 , 21)
6
30°
(− 1, 0) π 180°
210°
330°
1 cot2 x csc2 x
sin 2u 2 sin u cos u
cos 2u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u
2 tan u
tan 2u 1 tan2 u
Power-Reducing Formulas
1 cos 2u
2
1 cos 2u
2
cos u 2
1
cos
2u
tan2 u 1 cos 2u
sin2 u Sum-to-Product Formulas
2 x cos x
csc x sec x
2
sec x csc x
2
sin u sin v 2 sin
2 x sin x
tan x cot x
2
cot x tan x
2
cos
Reduction Formulas
sinx sin x
cscx csc x
secx sec x
x
(− 23 , − 12) 76π 5π 225°240° 300°315°7π 116π ( 23 , − 21)
(− 22 , − 22 ) 4 43π 270° 32π 53π 4 ( 22 , − 22 )
1
3
(0, − 1) ( 2 , − 2 )
(− 12 , − 23 )
Cofunction Identities
sin
0° 0
360° 2π (1, 0)
Double -Angle Formulas
1
tan x cot x
1
cot x tan x
Tangent and Cotangent Identities
tan x y
cosx cos x
tanx tan x
cotx cot x
Sum and Difference Formulas
sinu ± v sin u cos v ± cos u sin v
cosu ± v cos u cos v sin u sin v
tan u ± tan v
tanu ± v 1 tan u tan v
© Houghton Mifflin Company, Inc.
u 2 v cosu 2 v
uv
uv
sin u sin v 2 cos
sin
2 2 uv
uv
cos u cos v 2 cos
cos
2 2 uv
uv
cos u cos v 2 sin
sin
2 2 Product-to-Sum Formulas
1
sin u sin v cosu v cosu v
2
1
cos u cos v cosu v cosu v
2
1
sin u cos v sinu v sinu v
2
1
cos u sin v sinu v sinu v
2