ALGEBRA
Factors and Zeros of Polynomials
Let p(x) = a11x" + a, _ 1x" - 1 + · · · + a 1x + a 0 be a polynomial. If p(a) = 0, then a is a zero of the
polynomial and a solution of the equation p(x) = 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 p(x)
= ax2 + bx + c, and 0
::5 b 2 -
4ac, then the real zeros of pare x = ( - b ± v'b2
-
4ac)/2a.
Special Factors
x2
-
a2
x3 + a3
= (x -
a)(x + a)
= (x + a)(x2
x3
-
a 3 = (x - a)(x2
+ ax + a 2)
x4 - a4 == (x2 - a2)(x2 + a2)
-ax+ a 2)
Binomial Theorem
+ y )2 = x 2 + 2xy + y 2
(x + y )3 = x 3 + 3x2y + 3xy2 + y3
(x + y)4 = x 4 + 4x 3y + 6x2y 2 + 4xy3 + y4
(x - y )2
(x
(x + y)" = x" + nx" - 1y +
(x- y)" = x" - nx" - 1y +
n(n - I)
2!
n(n - I)
2!
=xz -
(x - y )3 =
x3 -
(x - y)4 = x4 -
+ y2
+ 3xy2 - y 3
4xly + fu2y2- 4xy3 + y4
2xy
3x 2y
~
x" - -yl + · · · + nxy" - 1 + y"
x" - 2 y 2
• • •
-
± nxy" - 1
::;::
y"
Rational Zero Theorem
If p(x) = anx" + a 11 _ 1x" - I + · · · + a 1x + a 0 has integer coefficients, then every
rational zero of p is of the form x = r/ s, where r is a factor of a 0 and s is a factor of a11•
Factoring by Grouping
acx3 + adx2 + bcx + bd
= ax2(cx +d) + b(cx +d) = (ax 2 + b)(cx +d)
Arithmetic Operations
ab
+ ac =
a
a(b + c)
c
b+d=
ad+ be
bd
m(~)(~) ~
a
-=-
a(~)= a:
a-b b-a
c-d=d-c
=
(~)
=
c
be
ab
+ ac
= b
+c
a
Exponents and Radicals
a0 =
I, a*O
(~Y = ~
(ab)X = aXbX
~
= am/n
axa>' =
a.r+y
1
a-x=ax
Ja = al/2
-a" = ax-y
~=~~
(UX)Y
a)'
= a-"Y
.zy'Q =
0 1/n
~=~
----
~~----'"1
FORMULAS fRQM GEOMETRY
Sector of Circular Ring
(p = average radius,
w = width of ring,
Triangle
h "" a sin ()
1
Area ""' - bh
2
(Law of Cosines)
c2
= a2
+
b2
-
Area = 8pw
2ab cos ()
Right Triangle
Ellipse
(Pythagorean Theorem)
cZ = a Z + bZ
Area = 1rab
b
Circumference "'" 27T
Jo/-
Cone
Equilateral Triangle
h=
w
()in radians)
~s
(A = area of base)
Ah
Volume = 3
2
~sz
Area = - 4
Parallelogram
Right Circular Cone
Area = bit
1rr2h
Volume = - 3
Lateral Surface Area ""
Frustum of Right Circular Cone
7r(r 2 + rR + R 2 )1t
Trapezoid
h
Area "" 2 (a + b)
Circle
Circumference "" 21Tr
Vo 1ume =
3
Lateral Surface Area = 1rs(R + r)
G
Right Circular Cylinder
Volume
= 1Trh
Lateral Surface Area
Sector of Circle
Sphere
(8 in radians)
Volume = i 1Tr3
3
Surface Area = 47rr2
()r2
Area = 2
s
=
w.J ~ + 112
r
o=
27Trh
co
c:
·e
Ill
Ill
.....
r()
Ill
Circular Ring
Wedge
(p = average radius,
w == width of ring)
Area = 7r(R 2 - r 2 )
= 21Tpw
(A ::r area of upper face,
B = area of base)
A = B sec()
~------------------------------------~------------------------------------~
co
ra
co
r::
~
cU
8.....
"'
-""
0
e
m
g