1. Topic: FUNDAMENTALS OF ALGEBRA
ALGEBRA
Algebra Properties
Property Illustration
Commutative property of addition a+b=b+a 2x + 3y = 3y + 2x
Commutative property of multiplication ab = ba (4 – x)y2 = y2(4 – x)
Associative property of addition (a + b) + c = a + (b + c) (4 + x) + x2 = x + (4 + x2)
Associative property of multiplication (ab)c = a(bc) (4x * 2)(8) = (4x)(2 * 8)
a(b + c) = ab + ac 5x(y + 3) = 5xy + 15x
Distributive properties
(a + b)c = ac + bc (5x + y)3 = 5xy + 15x
Additive identity property a+0=a 3x + 0 = 3x
Multiplicative identity property a*1=a 3x2 * 1 = 3x2
Additive inverse property a + (-a) = 0 3x2 + (-3x2) = 0
1 1
Multiplicative inverse property a* = 1 ; where a ≠ 0 (2x 2 + 4) * =1
a (2x 2 + 4)
Laws of Exponent
Rule Illustration
Rules of 1 x1 = x 41 = 4
Product rule x mxn = xm + n 32 * 35 = 37
mn m n
Power rule x = (x ) (82 )3 = 8 6
m
x 35
Quotient rule = x m − n ; where x ≠ 0 = 35 − 2 = 33
xn 32
Zero rule x 0 = 1 ; where x ≠ 0 50 = 1
1 1
Negative exponents x −n = 5 −1 =
xn 5
Laws of Radicals
For simplicity, rewrite radical expressions to exponents by applying the rule:
x
y
ay = ax
Illustration:
3x 2 y + 12x 4 y
(3x 2 y)1 / 2 + (12x 4 y)1 / 2
(31/2
) (
xy1 / 2 + 31 / 2 * 2x 2 y1 / 2 )
2
x 3y + 2x 3y
2
(x + 2x ) 3y
Infinity
Infinity, denoted by ∞ is sometimes used as a number, but it’s not really a number. It is simply a symbol that represents large
quantities
a
= ±∞
0
Note that the inverse of a small number is a large number.
Infinity rules do not obey the usual arithmetic rules, examples are:
∞+2 = ∞
∞ −3 = ∞
2*∞ = ∞
Indeterminate
“Indeterminate” is a term used when the mathematical expression is not definitely or precisely determined.
7 indeterminate forms:
(1) 00 (4) ∞−∞
0 ∞
(2) (5)
0 ∞
(3) 1∞ (6) 0*∞
(7) ∞0
Factoring
Difference of squares: a2 − b 2 = (a − b)(a + b) Factoring by grouping: 12xy + 3zx + 9zy + 4x 2
Common factor: 3a2b 2 − 6a2b + 6a2 = 3a2 (b2 − 2b + 2) 4x(x + 3y) + 3z(x + 3y)
Difference of cubes: 3 3 2
a − b = (a − b)(a + ab + b ) 2 (4x + 3z)(3y + x)
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2. Topic: FUNDAMENTALS OF ALGEBRA
Complex Numbers
i= −1
i2 = −i
i3 = −i
i4 = 1
3
Note: odd root is imaginary − 1 = −1
Rules of Inequality
Illustration:
4 − 3x ≤ −5 2 > −3 − 3x ≥ −7
− 3x ≤ −5 − 4 2 + 3 > −3x ≥ −7 + 3
x≥3 5 4
− <x≤
Equality is true, when x is greater than or 3 3
equal 3.
Functions
Functions are defined as a relation such that no two distinct pairs have the same first coordinate.
Relation is a set of distinct pairs (x, y).
Domain is the set of all first coordinates which gives a real value for y.
Range is set of all second coordinates.
Relation: {(0, 2), (4, 23), (90, 35)}
Domain: 0, 4, 90
Range: 2, 23, 35
Functional Notation:
Evaluate f(x) = x2 + x − 2 when f(0) and f(a)
1. f(0) = -2
2. f(a) = a2 – a – 2
Determinant
Determinants are distinct number associated with a square matrix. It is a mathematical item that are useful in the analysis
and solution of systems of linear equations (such as in Cramer’s rule), denoted by |A|.
Consider a square matrix shown below:
a b c
A= d e f
g h i
|A|= (aei + bfg +cdh) – (afh + bdi + ceg)
Note:
It is important to be familiar with the calculator that you are using, especially with its features and other capabilities. I personally
recommended to use CASIO fx-991ES.
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