Elgebra Math 10

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Real number
Irrational numbers π , √��

Rational numbers Integers Whole
Natural

3 5 1 2 4 2 2 3

Rational Like:

Integers {…, -3, -2, -1, 0, 1, 2, 3…….} Whole {0, 1, 2, 3…} Natural {1, 2, 3…}

, , ,

Properties of real numbers 1234-

Reflexive property a=a Symmetric property a = b then b = a Transitive property a = b and b = c then a = c Principle of substitution if a = b then we can substitute b for a in any expirations

Commutative properties a+b=b+a , a.b=b.a Associative properties a+(b+c)=(a+b)+c=a+b+c a.(b.c)=(a.b).c=a.b.c Distributive properties a.(b+c)=a.b+a.c (a+b).c=a.c+b.c Identity Properties 0+a=a+0=a a.1=a.a=a additive inverse Properties a + (- a ) = - a +a = 0 Multiplicative inverse properties a. = 1 �� 1 ��

Multiplication by zero a.0=0

. �� = 1 if b ≠ 0

0 ��

Division properties =0
�� ��

= 1 if a ≠ 0
�� −�� −�� �� �� �� −�� −�� �� ��

Rules of signs a(-b ) = - (ab) Exponents ��n = a.a.a…….a , (-a)b = - (ab) , ( -a ) ( -b ) = ab , - ( -a ) = a , ��0 = 1 if a ≠ 0 , ��−n = 1 �� n

=

=-

,

=

n factors ,

,

if a ≠ 0
1

Laws of exponents

��n ��m = ��m+n , (��m )n = ��mn Square roots √��2 =|��|

(����)n = ��n �� n ,

�� �� �� ��

= ����−�� =

�� ��−��

���� �� ≠ 0

,

( )�� =
�� ��

�� �� ���� ,

���� �� ≠ 0

Geometry Review

�� 2 = ��2 + �� 2

Pythagorean Theorem

Geometry Formulas
1

Area = LW Perimeter = 2L + 2W

Area = 2bh

Circumference = 2πr = πd

Area = π�� 2

Volume = LWH Surface area= 2LW+ 2LH+2WH

Volume= π�� 2 ℎ =π�� 2 ℎ + 2πrℎ Surface area=

Volume= 3 ���� 3
4

Surface area=4π�� 2

Polynomials
Special Products Difference of two squares

( �� + �� )2 = �� 2 + 2���� + ��2 ( �� − �� )2 = �� 2 − 2���� + ��2

( x – a )( x + a ) = �� 2 − ��2

Squares of binomials or perfect squares

( �� + �� )3 = �� 3 + 3���� 3 + 3��2 �� + ��3 ( �� − �� )3 = �� 3 − 3���� 3 + 3��2 �� + ��3 Differences of two cubes (�� − ��)(�� 2 + ���� + ��2 ) = �� 3 − ��3

Cubes of binomials or perfect Cubes

Sum of two cubes

(�� + ��)(�� 2 − ���� + ��2 ) = �� 3 + ��3

Summary Type of polynomial Any polynomial

Binomials of degree 2 or higher

method Look for common monomial factors (Always do this first!) Check for special product Difference of two squares Differences of two cubes

Trinomials of degree 2 Three or more terms
Nth Root
�� �� ��

Sum of two cubes Check for perfect square Grouping

�� 2 − ��2 �� 3 − ��3 �� 3 + ��3 ( �� ± �� )2

example 6�� 2 +9x = 3x(2x+3) ( x – a )( x + a )
(�� − ��)(�� 2 + ���� + ��2 ) (�� + ��)(�� 2 − ���� + ��2 ) �� 2 + 8�� + 16 = ( �� + 4 )2 �� 2 − �� − 2 = ( �� − 2 )( �� + 1 ) 6�� 2 + �� − 1 = ( 2�� + 1 )(3�� − 1 ) 2�� 3 − 3�� 2 + 4�� − 6 = (2�� − 3)(�� 2 + 2)

√���� = �� ���� �� ≥ 3 ���� ������ √���� = √�� √�� �� �� ��

Properties of radicals
��

√���� = |��| ���� �� ≥ 2 ���� ��������

√�� = �� ���������� �� = �� ��

��

Rational Exponents ��
1� ��

A�� 2 + bx + c = 0

Quadratic equation

= √��

��

��⁄ ��

= √���� =
��

� √�� �
��

��� = ��
��
��

√�� √��

��

��

√���� =

� √�� �
��

��

�� =

Quadratic Formula −�� ± √�� 2 − 4���� 2��

Discriminant of QUADRATIC Equation �� 2 − 4���� = ( > 0 ) or ( = 0 ) or ( < 0 ) Distance Formula

Line formula

D (P1, P2) =�(��2 − ��1)2 + (��2 − ��1)2 MIDPOINT FORMULA M = (X, Y) = ( ��1+��2 ��1+��2 , ) 2 2

Circles

STANDARD FORM OF AN EQATION OF A CIRCLE (X-H) 2 +(Y-K) 2= R2 X2+Y2+AX+BY+C=0

GENERAL FORM OF THE EQUATION OF A CIRCLE R = �(�� − ℎ)2 + (�� − ��)2 Slop of line M= x =a ��2− ��1 ��2 −��1

Lines

EQUATION OF THE VERTICAL LINE

POINT-SLOP FORM OF AN EQUATION OF A LINE Y-Y1=M(X-X1) Y=B SLOP-INTERSEPT FORM OF AN EQUATION OF A LINE Y=MX+B GENERAL FORM OF THE EQUATION OF THE LINE

EQUATION OF THE HORIZENTAL LINE

AX+BY=C A linear equation in one variable Ax + b = 0...
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