Document related concepts

Factorization wikipedia , lookup

Transcript
```FINAL EXAM REVIEW
JAMES mccroy
Algerbra-hartdke
14 may 2010
Example:
If a=b then a+c =b+c
Multiplication Property (of
Equality)
Example:
If a = b then a·c = b·c.
Reflexive Property (of
Equality)
Example:
a=a
Symmetric Property (of Equality)
Example:
If a=b then b=a
Transitive Property (of Equality)
Example:
If a=b and b=c then a=c
Associative Property of
Example:
c+(a+b)=a(c+b)
Associative Property of
Multiplication
Example:
c(ab)=a(cb)
Commutative Property of
Example:
a+b=b+a
Commutative Property of
Multiplication
Example:
a•b =b• a
Distributive Property (of
Example:
a(b+c)
ab+ac
Prop of Opposites or Inverse
Example:
11+ -11=0
Prop of Reciprocals or
Inverse Prop. of
Multiplication
Example:
1
6• =1
6
Identity Property of
Example:
a+0=a
Identity Property of
Multiplication
Example:
a•1= a
Multiplicative Property of Zero
Example:
a• 0 = 0
Closure property of real number addition states that
the sum of any two real numbers equals another real
number.
Example 2+5 = 7
Closure
Property
of
Multiplication
:
Closure property of real number multiplication states that the product of
any two real numbers equals another real number.
Example:
2 • 7 = 14
Product
of
Powers
Property
This property states that to multiply powers having the same base, add
the exponents.
Example:
an • bm = abn+m
Power
of
a
Product
Property
This property states that the power of a product can
be obtained by finding the powers of each factor and
multiplying them.
Example:
(3t)4 = 34 · t4 = 81t4
Power of a Power Property
This property states that the power of a power can be found by multiplying
the exponents.
Example:
(am )n = am•n
Quotient of Powers Property
This property states that to divide powers having the
same base, subtract the exponents.
Example:
a4
4-3
=
a
a3
Power of a Quotient Property
This property states that the power of a quotient can
be obtained by finding the powers of numerator and
denominator and dividing them.
Example:
(a ÷ b)2 = a2 ÷ b2
Zero Power Property
If any number is raised to the 0 power the answer is
automatically 0
0
Example: a =1
Negative Power Property
If any number is raised to a negative power the
Example:
1
a = b
a
-b
Zero Product Property
If the product of two or more factors is zero, then at
least one of the factors must be zero
Example:
If XY = 0, then X = 0 or Y = 0 or both X and Y are 0.
Product of Roots Property
The product of the roots of a quadratic equation is equal to
the
constant term divided by the leading coefficient.
Example:
 1 3
(-3)   = 2 2
Quotient of Roots Property
The square root of the quotient is the same as the quotient of the square roots
Example :
a
a
=
b
b
Root of a Power Property
Example:
a
2
Power of a Root Property
Example:
 a
2
Now you will take brief a quiz!
Look at the sample problem and give
the name of the property illustrated.
1. a + b = b + a
Commutative Property (of
Brief Quiz Cont…
2. If a = b then a·c = b·c
Brief Quiz Cont…
3. a(bc)=(ab)c
Associative Property of
Multiplication
Brief Quiz Cont…
4. 6(c+d)=6c+6d
Distributive Property of
Brief Quiz Cont…
5.
a
2
1
 2
a
Negative Power Property
Brief Quiz Cont…
5
a
2
6. 3  a
a
Quotient of a Powers Property
Brief Quiz Cont…
7. ab=ba
Commutative Property of
Multiplication
Brief Quiz Cont…
8.
a
a
=
b
b
Quotient of Roots Property
Brief Quiz Cont…
9.
a•1= a
Identity Property of Multiplication
Brief Quiz Cont…
10.
2 • 7 = 14
Closure Property of Multiplication
1st Power with Only One
Inequality Sign
11  x
0

11
1st Power- Conjunction
and




The word conjunction means there are two conditions in a
statement that must be met.
The greater than or less than signs will always be pointing
in the same direction
Look out for statements that cannot be true, such as the
following: 10 < x < 5
Always uses the word “and”
1st Power- Disjunction
or


Work as two separate inequalities
Always uses the word “or”
Slopes of All Types of Lines

Positive slope (when lines go uphill from left to right)

Negative slope (when lines go downhill from left to
right)

Undefined slope (when lines are vertical)

Zero slope (when lines are horizontal)
Equations of All Types of Lines

Standard/General Form
• Ax + By = C, where A > 0 and, if possible, A,
B, and C are relatively prime integers

Point-Slope Form
• y=mx+b or f(x)=mx+b




(synonyms)
The graph of this equation is a straight line
The slope of the line is m
The line crosses the y-axis at b
The point where the line crosses the y-axis is called
the y-intercept
Inconsistent System
4

2
Parallel lines don’t
intersect
• Null set
-2
-4
Consistent System
4
2
-2
-4

Line intersect at
one point
Dependent System
4

2
Infinite Set or All
Pts on the Line
• same line is used
twice
-2
-4
• Is consistent but
supersedes it
Linear Systems-Substitution Method

Replace one variable with an equal expression

Steps:
• Look for a variable with a coefficient of one.
• Isolate That variable
• Substitute this expression into that variable in Equation
B
• Solve for the remaining variable
• Back-substitute this coordinate into Step 2 to find the
other coordinate
Method (Elimination)


Combine equations to cancel out one
variable
Steps:
• Look for the LCM of the coefficients on either x
or y
• Multiply each equation by the necessary factor
• Add the two equations if using opposite signs
(if not, subtract)
• Solve for the remaining variable
• Back-substitute this coordinate into any
equation to find the other coordinate (Look for
easiest coefficients to work with)
Factoring Methods-GCF

Used for any # of terms

Factor GCF of equation

DONE
Factoring Methods-Sum/Diff of
Cubes



Used for binomials
The opposite of the
product of the cube roots
DONE
Factoring Methods-PST

Used for trinomials
If 1st & 3rd terms are
squares and the middle term
is twice the product of their
square roots
 DONE

Factoring Methods-Reverse FOIL

Trinomials

Trial and Error

DONE
Factoring Methods-Grouping

4 or more terms
• 2X2 Grouping



Look for two small
factorable groups
(glob) +PST
Pull the final GCF out in front of the leftover factors
DONE
2
• 3X1 Grouping



PST + perfect square
Write as (glob)2 +PST
DONE
9x2 – 30x + 25
a3 - b3
5x3 – 10x2 – 5x
9x2 – 30x + 25
75x4+108y2
6x2 – 17x + 12
Functions




F(x) is a synonym for the variable y
The domain of a function is the set of all possible x values
which will make the function "work" and will output real yvalues
The range of a function is the complete set of all possible
resulting values of the dependent variable of a function,
after we have substituted the values in the domain
The graph of a quadratic function is a curve called a
parabola. Parabolas may open upward or downward and
vary in "width" or "steepness" but they all have the same
basic "U" shape. All parabolas are symmetric with respect
to a line called the axis of symmetry. A parabola intersects
its axis of symmetry at a point called the vertex of the
parabola.
Simplifying Expressions with
Exponents


To simplify with exponents, you
DON’T have to work only from the
rules for exponents. It is often easier
to work directly from the definition
and meaning of exponents
Keep in mind The Negative Power
Property and The Zero Power
Property
Simplifying Expressions With

Break down a number into its
smaller pieces, you can do
the same with variables until
the radical is a square root
WORD PROBLEMS!!!!




Eleven cards and three boxes of candy cost \$35.39. Twelve
cards and four boxes of candy cost \$42.68. How much do
two cards cost?
Lois rides her bike to visit a friend, She travels at 10mph
while she is there it starts to rain. Her friend drives her
home in a car traveling at 25 mph. It takes Lois 1.5 hours
longer to go to her friends house than it does for her to
return home. How many hours did it take to ride to her
friends house?
Al's father is 45. He is 15 years older than twice Al's age.
How old is Al?
Karen is twice as old as Lori. Three years from now, the
sum of their ages will be 42. How old is Karen?
Line of Best Fit or Regression Line



Is the BEST process of constructing a
curve, or mathematical function, that
has the best fit to a series of data
points, possibly subject to limitations
Easy to do on a graphing calculator
because it does it automatically
EXAMPLE:
```