Prime Factors of 12 : 2, 3
(See also prime numbers.)

To factor is to find all of the factors of an expression or a number.
Examples:
factor 50:
50 = 2 5 5
factor x³ + 4x² +3x:
x³ + 4x² +3x = x(x + 1)(x + 3)

Some special factors:
a² ± 2ab + b² = (a ± b)²
a³ ± 3a²b + 3ab² ± b³ = (a ± b)³
a² - b² = (a - b)(a + b)
a³ ± b³ = (a ± b)(a² ± ab + b²)

Area

of a square	A = s2			s = side l = length w = width b = base h = height r = radius
of a rectangle	A = lw
of a triangle	A =	1 bh 2
of a circle	A = r2

Perimeter

of a square	P = 4s
of a rectangle	P = 2l +2w

Circumference

of a circle

C = 2r

Volume

of a cube	V =	s3		s = side l = length w = width b = base h = height r = radius
of a box	V =	lwh
of a cylinder	V =	r2h
of a cone	V =	1r2h 3
of a sphere	V =	4r3 3

Examples:

Fractions:	1,   2	a,   b	4,   d	x + 3 y
Numerators:	1,	a,	4,	x + 3
Denominators:	2,	b,	d,	y	(b, d, y 0)

When the denominator of a fraction is zero, the fraction is undefined.

Equivalent Fractions:	1 2	=	3 6	=	2x 4x	=	x + 5  2(x + 5)
1  is in the lowest terms among its equivalent fractions. 2

Improper Fractions:

5,   3

9,   4

26,    7

8 2

An improper fraction is a fraction with numerator denominator. It may be converted to a mixed number or a whole number:

1

2,   3

2

1,   4

3

5,   7

4

Proper Fractions:

4,   5

1,   6

7,   9

11 24

A proper fraction is a fraction with numerator < denominator.

Complex Fractions:

2    3    4 5

,

x + 1  x 3

,

2    9    5

,

2 + 1      x      x

A complex fraction is a fraction whose numerator or denominator is also a fraction. It may be converted to a simple fraction:

10 12

,

3(x + 1) x

,

2   45

,

2x + 1 x2

Examples:
y = 3x + 1
y = sin x

x is called the independent variable and y as a function of x, is called the dependent variable. The set of all the x values that makes the function meaningful is called the domain and the set of all the possible y values of the function is called the range.

Thus, the domain of y = 3x +1 is the real line and the range of y = 3x +1 is the real line.

The domain of y = sin

1 x

is all real x except x = 0 and the range is -1 < y < 1.

G.C.F. of (9, 15) = 3
since 9 = 3 3 and 15 = 3 5

G.C.F. of (7, 6) = 1
since 7 = 7 and 6 = 2 3
When there is no common factor then G. C. F. is 1.

The square root of a negative number is an imaginary number. It can be expressed in the form
ai where a is a real number and i =

Examples:
2i, -3.4i, ( = i ) are imaginary numbers.
(See also number and complex number.)

4 is greater than 3	4 > 3
x is greater than or equal to 7	x 7
2 is less than 5	2 < 5
y is less than or equal to 1	y 1

We may use these properties to solve inequalities like we solve equations.

Solve:  2x > 5x + 6
-3x > 6
   x < -2 (solution)

L.C.M. of (6, 9) = 18
since 6 = 2 3 and 9 = 3 3
L.C.M. = 2 3 3 = 18

Find the L.C.M. of (7, 15, 21)
since 7 = 7, 15 = 3 5 and 21 = 3 7
L.C.M. = 7 3 5 = 105

Example:
Find the least common denominator (L.C.D.) of:

1 2	,	3 4	,	1 6
The L.C.D. of	1 2	,	3 4	,	1 6	= L.C.M. of (2, 4, 6) = 12.

A logarithmic function is a function of the form:
y = log_ax where a > 0 and a 1
and x and y are related as:
a^y = x

Some properties of logarithms:
log_a1 = 0
log_a(xz) = log_ax + log_az

loga(

x z

) = logax - logaz

log_axⁿ = n log_ax

(See also exponential function and function.)

Examples:

3

2 7

,

1

4 11

,

2

1 2

are mixed numbers.

(See also fraction.)