Download Algebra Factoring Formulas Product Formulas

Algebra
Factoring Formulas
Real numbers : a, b, c
Natural number : n
1. a2 – b2 = ( a + b)(a – b )
2. a3 – b3 = ( a – b )(a2 + ab + b2)
3. a3 + b3 = ( a + b )(a2 - ab + b2)
4. a4 – b4 = (a2 – b2)( a2 + b2) = ( a + b)(a – b ) )( a2 + b2)
5. a5 – b5 = ( a – b)(a4 + a3b + a2b2 + ab3 + b4)
6. a5 + b5 = ( a + b)(a4 - a3b + a2b2 - ab3 + b4)
7. If n is odd, then
an + bn = ( a + b)(an-1 – an-2b + an-3b2 - … - abn-2 + bn-1).
8. If n is even, then
an – bn = ( a - b)(an-1 + an-2b + an-3b2 + … + abn-2 + bn-1).
an + bn = ( a + b)(an-1 – an-2b + an-3b2 - … + abn-2 - bn-1).
Product Formulas
Real numbers : a, b, c
Whole numbers : n, k
9.
( a – b )2 = a2 – 2ab + b2
10.
( a + b )2 = a2 + 2ab + b2
11.
( a – b )3 = a3 – 3a2b + 3ab2 – b3
12.
( a + b )3 = a3 + 3a2b + 3ab2 + b3
13.
( a – b )4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
14.
( a + b )4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
15.
Binomial Formula
( a + b )n = nC0an + nC1an-1b + nC2an-2b2 + … + nCn-1abn-1 + nCn-1bn,
16.
( a + b + c )2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
17.
( a + b + c + … + u + v )2 = a2 + b2 + c2 + … + u2 + v2 +
+ 2 ( ab + ac + … + au + av + bc + … + bu + bv + … + uv )
Powers
Bases (positive real numbers) : a, b
Powers (rational numbers) : n, m
18.
aman = am + n
19.
= am – n
20.
( ab )m = ambm
21.
(
22.
(am)n = amn
23.
a0 = 1, a
24.
a1 = 1
25.
a-m =
)m =
0
26.
=
Roots
Bases : a, b
Powers (rational numbers) : n, m
a, b
0 for even roots ( n = 2k, k
27.
=
28.
=
29.
=
30.
=
,b
=
31.
(
)p =
32.
(
)n = a
33.
=
34.
=
35.
=
36.
37.
(
0
,b
)m =
=
,a
0.
0
N)
38.
=
39.
=
Logarithms
Positive real numbers : x, y, a, c, k
Natural number : n
40.
Definition of Logarithm
if and only if x =
y=
41.
=0
42.
=1
43.
=-
if a > 1 and +
44.
=
+
45.
=
-
46.
=n
47.
=
48.
=
49.
=
50.
x=
=
, a > 0, a
1.
if a < 1
, c > 0, c
1.
51.
Logarithm to Base 10
= log x
52.
Natural Logarithm
= ln x,
= 2.718281828…
where e =
=
53.
54.
ln x =
1n x = 0.434294 1n x
logx = 2.302585 log x
Equations
Real numbers : a, b, c, p, q, u, v
Solutions : x1, x2, y1, y2, y3
55.
Linear Equation in One Variable
ax + b = 0 , x = -
56.
Quadratic Equation
ax2 + bx + c = 0, x1,2 =
57.
Discriminant
D = b2 – 4ac
58.
Viete’s Formulas
If x2 + px + q = 0, then
x1 + x2 = -p and
x1x2 = q
59.
ax2 + bx = 0, x1 = 0, x2 = -
60.
ax2 + c = 0, x1,2 =
61.
Cubic Equation. Cardano’s Formula.
y3 + py + q = 0,
y1 = u + v, y2,3 = -
( u + v)
( u +v )i,
where
u=
,v=
Inequalities
Variables : x, y, z
Real numbers : a, b, c, d, a1, a2, a3 …,an, m, n
Determinants : D, Dx, Dy, Dz
62.
Inequalities, Interval Notations and Graphs
a
Inequality
x b
Interval Notation
[a, b]
a
x
b
(a, b]
a
x
b
[a, b)
Graph
a
x
x
b
(a, b)
x
b,
(-
, b]
x
b,
(-
, b)
b
x
b
a
x
x
a
,
[a,
)
a
x
x
a
,
(a,
)
63.
If a > b, then b < a.
64.
If a > b, then a – b > 0 or b – a < 0.
65.
If a > b, then a +c > b + c.
66.
If a > b, then a – c > b – c.
67.
If a > b and c > d, hen a + c > b + d.
68.
If a > b and c > d, hen a - d > b - c.
69.
If a > b and c > d, then ma > mb.
70.
If a > b and m > 0, then
71.
If a > b and m > 0, then ma < mb.
72.
If a > b and m < 0, then
73.
If 0 < a < b and n > 0, then an < bn .
74.
If 0 < a < b and n < 0, then an > bn .
75.
If 0 < a < b, then
76.
<
>
<
.
.
.
,
where a > 0, b > 0; an equality is valid only if a = b.
77.
a+
2, where a > 0 ; an equality takes place only at a = 1.
, where a1, a2, ... , an > 0.
78.
79.
If ax + b > 0 and a > 0, then x > -
.
80.
If ax + b > 0 and a < 0, then x < -
.
81.
ax2 + bx + c > 0
a>0
a<0
D>0
x
< x1, x > x2
x1 < x < x2
D=0
x
x1 < x, x > x1
D<0
-
82.
<x<
x
+
83.
If
< a, then –a < x < a , where a > 0.
84.
If
< a, then x < -a and x > a, where a > 0.
85.
If x2 < a, then
<
, where a > 0.
86.
If x2 > a, then
>
, where a > 0.
87.
If
> 0 , then f(x) . g(x) > 0 and g(x)
0
88.
If
< 0 , then f(x) . g(x) < 0 and g(x)
0
Compound Interest Formulas
Future value : A
Initial deposit : C
Annual rate of interest : r
Number of years invested : t
Number of times compounded per year : n
89.
General Compound Interest Formula
A=C(1+
90.
)nt
Simplified Compound Interest Formula
If interest is compounded once per year, then the previous formula simplifies
to :
A = C ( 1 + r )t .
91.
Continuous Compound Interest
If interest is compounded continually ( n
A = Cert .
), then