- The document contains information on algebraic rules and properties including: the quadratic formula, factoring expressions, operations with exponents, logarithmic rules, and common algebraic errors to avoid.
- It provides examples of applying concepts like factoring trinomials, using the binomial theorem, and expanding expressions with exponents.
- Logarithmic rules covered include defining the logarithmic function, common and natural logarithms, and properties like changing bases.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
College Algebra Real Mathematics Real People 7th Edition Larson Solutions Manual
full download: https://goo.gl/ebHcPK
People also search:
college algebra ron larson 7th edition pdf
college algebra: real mathematics, real people pdf
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
College Algebra Real Mathematics Real People 7th Edition Larson Solutions Manual
full download: https://goo.gl/ebHcPK
People also search:
college algebra ron larson 7th edition pdf
college algebra: real mathematics, real people pdf
college algebra real mathematics answers
webassign
1. − ± 2 − 4
= .
x b b ac
− 3 ± 13
a + c
= ad +
bc
a b
=
b
a +
⎛
⎛
b a
a b
a b
ab ac a b c
= +
+ ( ) ,
1 ⎞
−
−
b c
⎛
−
−
ac
The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC
ad
a
1
Basic Rules for
Algebra
&
Logarithmic Functions
Quadratic Formula: Example:
If p(x) = ax2 + bx + c , a ≠ 0 and 0 ≤ b2 − 4ac ,
then the real zeroes of p are
a
2
If p(x) = x2 + 3x −1=0, with a=1, b=3, and c=-1, then p(x)
=0 if
2
x =
Special Factors: Examples:
x2 − a2 = (x − a)(x + a) x2 − 9 = (x − 3)(x + 3)
x3 − a3 = (x − a)(x2 + ax + a2 ) x3 − 8 = (x − 2)(x2 + 2x + 4)
x3 + a3 = (x + a)(x2 − ax + a2 ) x3 + 27 = (x + 3)(x2 − 3x + 9)
Binomial Theorem Examples:
(x + a)2 = x2 + 2ax + a2 (x + 3)2 = x2 + 2(x)(3) + 32 = x2 + 6x + 9
(x − a)2 = x2 − 2ax + a2 (x − 5)2 = x2 − 2(x)(5) + 52 = x2 −10x + 25
(x + a)3 = x3 + 3ax2 + 3a2x + a3 (x + 2)3 = x3 + 3(x)2 (2) + 3(x)(2)2 + 23 = x3 + 6x2 +12x + 8
(x − a)3 = x3 − 3ax2 + 3a2x − a3 (x −1)3 = x3 − 3(x)2 (1) + 3(x)(1)2 −13 = x3 − 3x2 + 3x −1
Factoring by Grouping Example:
acx3 + adx2 + bcx + bd = ax2 (cx + d) + b(cx + d) =
3x3 − 2x2 − 6x + 4 = x2 (3x − 2) − 2(3x − 2) =
(ax2 + b)(cx + d)
(x2 − 2)(3x − 2)
Arithmetic Operations:
ab + ac = a(b + c)
bd
d
b
+
c
c
c
bc
d
c
a
b
c
d
a
b
a
c b
d
= ÷ = × =
⎞
⎞
⎟⎠
⎟⎠
⎜⎝
⎜⎝
ab
c
⎟⎠
a ⎛
b ⎞
= a × b
= c
c
⎜⎝
1
d c
c d
c d
−
=
−
⎟⎠
⎜⎝
−
=
−
1
a
a
+
=
c a
a
a
b
⎞
⎛
a ≠ 0 c
b
b c
bc
= ÷ = × =
⎟⎠
⎜⎝
1
1
b
a = ÷ = a × c
=
b
a b
c
b
⎛ 1 1
c
⎞
⎟⎠
⎜⎝
2. x
⎛ a
n a m a n (n a
)m a = ⎟⎠
a = − x
a− = 1 (ax )y = axy 2
x
x
x
= a
= ⎛ 1
1
x
a
⎞
⎞
⎛
⎟⎠
a bx = + = +
x x b log
x x b ln
The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC
2
Exponents and Radicals
a0 = 1, a ≠ 0 x y
y
a
a
x x
b
b
⎞
⎜⎝
m = =
x
x
a
1 a = a n ab = n a ∗ n b
n
a = ⎟⎠
⎛
axa y = ax + y (ab)x = axbx n a = a1n n
n
a
b
b
⎞
⎜⎝
Algebraic Errors to Avoid:
a
b
a ≠ a
+
+
x
x b
1 ≠ .
To see this error, let a, b, and, x all equal to 1. Then, 2
2
x2 + a2 ≠ x + a To see this error, let x = 3 and a = 4. Then, 5 ≠ 7 .
a − b(x −1) ≠ a − bx − b Remember to distribute the negative signs. The equation should be
a − b(x −1) = a − bx + b
bx
a
x
a
b
≠
⎞
⎟⎠
⎛
⎜⎝
To divide fractions, invert and multiply. The equation should be
ab
a b
b
b
= ⎟⎠
⎜⎝
⎜⎝
− x2 + a2 ≠ − x2 − a2 We can’t factor a negative sign outside of the square root.
bx
a +
bx ≠ +
a
1
This is one of many examples of incorrect cancellation. By applying
the properties above, the equation should be 1 bx
.
a
bx
a
a
a
+
a
(x2 )3 ≠ x5 In this case we multiply the exponents, then the correct equation is
(x2 )3 = x2x2x2 = x2+2+2 = x6
LOGARITHMIC RULES
Logarithmic function y x a = log , defined by x = ay
Common logarithm x x 10 log = log
Natural logarithm x x e ln = log
Log Properties
ax x
a log = ; aloga x = x
(MN) M N a a a log = log + log
log (M / N) = log M − log
N a a a log (M) N
=
N(log M) a a
Change of base formulas
b
log = log
b
log = ln