Introductory Algebra - Reference Guides

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INTRODUCTORY ALGEBRA • A-571-11
TERMINOLOGY
Rational numbers Ratios of integers: 3/4, –8/5, 127/16,
–3/1 = –3
Irrational numbers Numbers that cannot
be expressed as ratios
of integers:
Algebraic expressions Combinations of
numbers and letters:
Terms of an Products or quotients of numbers and letters:
algebraic expression
is an algebraic expression, not a term
Monomial Algebraic expressions consisting of one term
Binomial Algebraic expressions consisting of two terms
Multinomial Algebraic expressions
consisting of more
than two terms:
Polynomial Algebraic expressions consisting of several
terms that are integral and rational in the
lettered terms: 5x46x3+ x22x + 1
but 3/x + y, are not polynomial
Degree of monomial Sum of exponents of lettered symbols:
4x3y2z5is of degree 10
Degree of polynomial Exponent of the term of the highest degree:
5x46x3+x2x + 1 has a degree of 4
INTEREST
P= principal
I= accumulated interest
A= principal + interest
R= interest rate per year
N= number of years
I= P ¥R ¥N
A= P(1 + RN)
Example
• Interest to be paid on
$4,000 borrowed at 10%
for 5 years
I = PRN = 4000 ¥0.10 ¥5
= $2,000
• Principal to be invested at
7% that will yield $10,000
in 5 years
71
323
,,,
/
π
e
56 2 5
26
2
3
ab xy
bc d
a
+
,,
51
251
2
2323
xy ab
cxy ab
c
, but
63
22
x
y
z
xxyz+ +
43z+
a, b, c, u, v, w
= real constants
A, B, C, D, E
= algebraic expressions
f(x), g(x)
= functions of x
SIMPLE INTEREST
PA
RN
=+=
=
1
10 000
1007 5
740741
,
.
$, .
P= principal
I= compound interest
A= compound amount
R= interest rate per period
N= number of interest periods
I= A P
A= P(1 + i)N
Example
• Compound amount Areceived
and compound interest Iearned
on principal of $5,000
compounded semi-annually at 5%
over 2 years
i= 0.05/2 = 0.025
N= 4
P= $5,000
A= P(1 + i)N= 5000 (1 + 0.025)4
= $5,519.06
I= A P = 5519.06 – 5000
= $519.06
COMPOUND INTEREST
QUADRAT I C EQUAT I O N S
• Every quadratic equation (ax2+ bx + c = 0)in one unknown
has exactly two roots (x1and x2) that can be real, equal, or
complex
Notes
• If one is complex, then the 2nd is its conjugate: A ± Bi
• The sum of the roots is given by: x1+ x2= –b/a
• The product of the roots is given by: x1x2= c/a
QUADRATIC FORMULA THE ROOTS
• The roots of a
quadratic equation are
given directly by the
relation:
xbb ac
a
12
24
2
,=−± −
Example
Given:
Substitute in:
and
3850
864453
23
153
2
12
12
xx
x
xx
++=
=−± − ××
×
=− =−
,
/
Nature of the Roots
The Discriminant, D = b24ac
D = b24ac > 0 Roots are
real and
unequal
D = b24ac = 0 Roots are
real and
equal
D = b24ac < 0 Roots are
complex
conjugates
D = 4ac < 0 Roots are
pure
imaginary
Introductory Algebra
Introductory Algebra
TM
permacharts
DEALING WI T H FRACTIONS
RULES EXAMPLE
Rule 1
The numerator/denominator
can be multiplied/divided by
the same factor
Rule 2
To add/subtract fractions,
write each fraction in terms
of a common denominator,
preferably the lowest
common denominator (LCD)
Rule 3
To multiply fractions, multiply
the numerators and
denominators separately
Rule 4
To divide by a fraction, invert
the fraction and multiply
Rule 5
To divide sums or differences
of fractions, reduce the
numerator and denominator
to single fractions using LCDs,
and then divide the result
x
y
x
y
x
y
xz
yz
=
==
3
3
/
/
72
1
31
71231
1
943
1
2
2
2
2
2
2
xx xLCD x x
xx x x
xx
xx
xx
==
=−+ + −
=−−
()
() ()
()
()
2
3
612
3
4
22
a
b
b
a
ab
ab a
×= =
a
aa
a
aa
aa
aa
aa
aa
aa aa
a
+
+
+−
+−
−+
−+ +
+−
+=
=++−+−
=
1
11
1
1111
11
11
11
11
22
22
21 21
22
()()
()()
()()
()()
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