Algebra 1 A Review and Summary Gabriel Grahek

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Algebra 1 A Review and SummaryGabriel Grahek
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In the next slides you will reviewSolving 1st
power equations in one variableA. Special cases
where variables cancel to get all reals or
B. Equations containing fractional
coefficientsC. Equations with variables in the
denominator (throw out answers that cause
division by zero)
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1st Power Equations

• Any type of equation that has only one variable.

• x38 4(x2-2)836
• x3-38 (4x2-8)836
• x5 4x2-828
• 4x236
• x29
• x3

Note that the variable can be on both sides.
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1st Power Equations

• 3x-412x 4y1624
• 3x-44124x 4y16-1624-16
• 3x16x 4y8
• 3x-x16x-x y2
• 2x16
• x8

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1st Power Equations

• Special cases- Ø and all reals

x-(4-3)x 4x42(2x2) x-1x 4x4x -10
xx Ø
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1st Power Equations

• Equations containing fractional coefficients and
  with variables in the denominator.

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In the next slides you will reviewReview all
the Properties and then take a Quiz on
identifying the Property Names
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Addition Property (of Equality)
Example If a, b, and c, are any real numbers,
and ab, then acbc and cacbIf the same
number is added to equal numbers, the sums are
equal.
Multiplication Property (of Equality)
Example If a, b, and c are real numbers, and
ab, then cacb and acbc. If equal numbers are
multiplied by the same number, the products are
equal.
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Reflexive Property (of Equality)
Example For all real numbers a, b, and caa
Symmetric Property (of Equality)
Example For all real numbers a, b, and cIf
ab, then ba.
Transitive Property (of Equality)
Example For all real numbers a, b, and cIf
ab and bc, then ac.
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Associative Property of Addition
Example For all real numbers a, b, and
c(ab)ca(bc) Example (56)75(67)
Associative Property of Multiplication
Example For all real numbers a, b, and
c (ab)ca(bc) Example
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Commutative Property of Addition
Example For all real numbers a and babba
Example 2332
Commutative Property of Multiplication
ExampleFor all real numbers a, b, and cabba
Example
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Distributive Property (of Multiplication over
Addition)
Example For all real numbers a, b, and
ca(bc)abacand(bc)abaca
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Prop of Opposites or Inverse Property of Addition
Example For every real number a, there is a
real number -a such thata(-a)0 and (-a)a0
Prop of Reciprocals or Inverse Prop. of
Multiplication
Example If we multiply a number times
its reciprocal, it will equal one. For example
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Identity Property of Addition
Example There is a unique real number 0 such
that for every real number a, a 0 a and 0 a
a Zero is called the identity element of
addition.
Identity Property of Multiplication
Example There is a unique real number 1 such
that for every real number a, a 1 a and 1 a
a One is called the identity element of
multiplication.
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Multiplicative Property of Zero
Example For every real number a, a 0 0 and
0 a 0
Closure Property of Addition
Example Closure property of real number
addition states that the sum of any two real
numbers equals another real number.
Closure Property of Multiplication
Example Closure property of real number
multiplication states that the product of any two
real numbers equals another real number.
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Product of Powers Property
Example This property states that to multiply
powers having the same base, add the
exponents.That is, for a real number non-zero a
and two integers m and n, am an amn.
Power of a Product Property
Example This property states that the power of
a product can be obtained by finding the powers
of each factor and multiplying them. That is,
for any two non-zero real numbers a and b and any
integer m, (ab)m am bm.
Power of a Power Property
Example This property states that the power of
a power can be found by multiplying the
exponents.That is, for a non-zero real number a
and two integers m and n, (am)n amn.
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Quotient of Powers Property
Example This property states that to divide
powers having the same base, subtract the
exponents.That is, for a non-zero real number a
and two integers m and n, .
Power of a Quotient Property
Example This property states that the power of
a quotient can be obtained by finding the powers
of numerator and denominator and dividing them.
That is, for any two non-zero real numbers a and
b and any integer m,
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Zero Power Property
Example Any number raised to the zero power is
equal to 1.
Negative Power Property
Example Change the number to its reciprocal.
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Zero Product Property
Example Zero - Product Property states that if
the product of two factors is zero, then at least
one of the factors must be zero. If xy 0, then
x 0 or y 0.
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Product of Roots Property
For all positive real numbers a and b, That
is, the square root of the product is the same as
the product of the square roots.
Quotient of Roots Property
For all positive real numbers a and b, b ? 0
The square root of the quotient is the same as
the quotient of the square roots.
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Root of a Power Property
Example
Power of a Root Property
Example
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
1. a b b a
Answer Commutative Property (of Addition)
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
2. am an amn
Answer Product of Powers
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
3. For every real number a, a 0 0 and 0 a
0
Answer Multiplicative Property of Zero
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
4. the sum of any two real numbers equals another
real number.
Answer Closure Property of Addition
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
5. There is a unique real number 1 such that for
every real number a, a 1 a and 1 a a
Answer Identity property of Multiplication
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
6.
Answer Zero Power Property
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
7.
Answer Quotient of Powers Property
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
8. (ab)m am
Answer Power of a Product
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
9.
Answer Negative Power Property
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
10.
Answer Prop of Reciprocals or Inverse Prop.
of Multiplication
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Now you will take a quiz!Look at the sample
problem and give the name of the property
illustrated.
Click when youre ready to see the answer.
11. (ab)ca(bc)
Answer Associative Property of Multiplication
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Solving Inequalities
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Solving Inequalities

• Remember the Multiplication Property of
  Inequality! If you multiply or divide by a
  negative, you must reverse the inequality sign.

-2x lt 8
x gt -4
Solution Set x x gt -4
Graph of the Solution
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Solving Inequalities

• Open endpoint for these symbols gt lt
• Closed endpoint for these symbols or
• Conjunction must satisfy both conditions
• Conjunction AND

x -5 lt x 8
Click to see solution graph
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Solving Inequalities

• Open endpoint for these symbols gt lt
• Closed endpoint for these symbols or
• Disjunction must satisfy either one or both of
  the conditions
• Disjunction OR

x x lt -6 or x 8
Click to see solution graph
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Solving Inequalities Special Cases

• Watch for special cases
• No solutions that work Answer is Ø
• Every number works Answer is reals
• When the disjunction goes the same way you use
  one arrow.

x x gt -6 or x 8
Click to see solution graph
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Solving Inequalities Special Cases

• Watch for special cases
• No solutions that work Answer is Ø
• Every number works Answer is reals

x -2x lt -4 and -9x 18
Click to see solution
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Solving Inequalities

• Now you try this problem

2x gt 6 or -16x 32
Click to see solution and graph
-2 lt x or x 3
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Solving Inequalities

• Now you try this problem.

4x-8 lt 12 and -x lt 10-4
Click to see solution and graph
-6 lt x lt 5
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Type the answer here. Set to fade-in on click
Type a sample problem here. Blah blah blah. You
can duplicate this slide.
Click when ready to see the answer.
Type any needed explanation or tips here. Set to
fade-in 3 seconds after the answer appears above.
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In the next slides you will reviewLinear
equations in two variables Lots to cover here
slopes of all types of lines equations of all
types of lines, standard/general form,
point-slope form, how to graph, how to find
intercepts, how and when to use the point-slope
formula, etc. Remember you can make lovely
graphs in Geometer's Sketchpad and copy and paste
them into PPT.
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Linear Equations

• Slope
• Point-Slope Formula
• Slope-Intercept Formula
• Midpoint Formula
• Standard/ General Form AxBxC
• Distance Between Two Points Formula

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Slope
Pt-Slope Formula
(9,12) and (13, 20) Use when you only have
solution points.

• (9,12) and (13, 20)
• Would be negative if it had a negative sign in
  front of it. It would then be a falling line and
  not a rising line.

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Midpoint
Distance
(9,12) and (13, 20) Use to find the Distance
between to points.

• (9,12) and (13, 20)
• Use to find the middle point on a line.

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Equations in Two Variables

• The pairs of numbers that come out for each
  variable can be written as an (x,y) value.
  (ordered pair)
• You give the solutions in alphabetical order of
  the variables. So, it would be (a,b) and not
  (b,a).

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Standard Form

• axbyc
• All linear equations can be written in this form.
• A, b, and c are real numbers and a and b are
  non-zero. A, b, and c are integers.
• To change to slope intercept
• Axbxc bxaxc

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How to graph

• To graph the slope-intercept form you can take
  the y intercept and use the slope to determine
  the points on the line.
• To graph the standard form you have to change it
  to slope-intercept, explained in the last slide,
  and then graph it.

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To find the y-intercept
To find the x-intercept
F(x)mxb Set f(x)0 0mxb Divide
out Example F(x)4x-8 04x-8 84x 2x

• F(x)mxb
• Set x to 0
• Fb
• Example
• F(x)4x6
• F(0)4(0)6
• F6

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In the next slides you will reviewLinear
Systems A. Substitution Method B.
Addition/Subtraction Method (Elimination ) C.
Check for understanding of the terms dependent,
inconsistent and consistent
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When two line share solution points

• Null set (if they are parallel)
• This will be called an INCONSISTENT SYSTEM
• One point (if they cross)
• This will be called a CONSISTENT SYSTEM
• Infinite Set or All Pts on the Line (if same line
  is used twice)
• This will be called a DEPENDENT SYSTEM (It is
  also consistent. Dependent is the better name
  for it than consistent.)

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• Solution Null set (if they are parallel)
• INCONSISTENT SYSTEM
• One point (if they cross)
• CONSISTENT SYSTEM
• Infinite Set or All Pts on the Line (if same line
  is used twice)
• DEPENDENT SYSTEM

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The SOLUTION of a SYSTEM is the INTERSECTION SET
Where do the two lines Meet intersect.
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Two Different Equation on the same graph are
called a SYSTEM OF EQUATIONS.

• Think about

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Method 1 To estimate the solution of a system,
you have to find out where they intersect.
Solution to this system is (2, 1)
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Example You can use Trace on a graphing
calculator to help you estimate the solution of a
system. It can find where they intersect.
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Example You can use Trace on a graphing
calculator to help you estimate the solution of a
system. It can find where they intersect
Solution to this system appears to include TWO
pts (-2.38, 1.79), (3, 0)
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Summary of Method 1 Estimate the SOLUTION of a
SYSTEM on a graph. (Goal Find intersection
pts.)
Disadvantages Might only give an estimate. It
might not be possible to graph some equations
yet. AdvantagesIf the graph is easy, this is a
good way to check. It is good for a quick answer.
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Method 2 Substitution Method(Goal Replace
one variable in one equation with the set from
another.)
Step 1 Look for a variable with a coefficient
of one. Step 2 Move everything else to the
other side. Equation A now becomes y
15-x Step 3 SUBSTITUTE this expression into
that variable in Equation B Equation B now
becomes 4x 3( 15-x ) 38 Step 4 Solve the
equation. Step 5 Back-substitute this coordinate
into Step 2 to find the other coordinate. (Or
plug into any equation but step 2 is easiest!)
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Method 2 Substitution (Goal replace one
variablewith an equal expression.)
Step 1 Look for a variable with a coefficient
of one. Step 2 Isolate that variable
Equation A now becomes y 3x
1 Step 3 SUBSTITUTE this expression into that
variable in Equation B Equation B becomes
7x 2( 3x 1 ) - 4 Step 4 Solve
for the remaining variable Step
5 Back-substitute this coordinate into Step 2 to
find the other coordinate. (Or plug into any
equation but step 2 is easiest!)
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Example Substitution (Goal replace one
variable with the set of another equation.)
Step 1 Look for a variable with a coefficient
of one. Step 2 Move everything else to the
other side. Step 3 SUBSTITUTE this
expression into that variable in Equation B
Step 4 Solve for the remaining variable Step
5 Back-substitute this coordinate into Step 2 to
find the other coordinate. (Or plug into any
equation but step 2 is easiest!)
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Example Substitution (Goal replace one
variablewith an equal expression.)
Step 1 Look for a variable with a coefficient
of one. Step 2 Isolate that variable Step 3
SUBSTITUTE this expression into that variable in
Equation B Step 4 Solve for the remaining
variable Step 5 Back-substitute this coordinate
into Step 2 to find the other coordinate. (Or
plug into any equation but step 2 is easiest!)
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Method 2 Summary Substitution Method(Goal
replace one variable with an equal expression.)
Disadvantages Avoid this method when it
requires messy fractions ? Avoid IF no
coefficient equals one. AdvantagesThis is
the algebra method to use when degrees of the
equations are not equal.
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Method 3 Elimination Methodor
Addition/Subtraction Method(Goal Combine
equations to cancel out one variable.)
Step 1 Look for the LCM of the coefficients on
either x or y. (Opposite signs are recommended
to avoid errors.) Here -3y and 2y could be
turned into -6y and 6y Step 2 Multiply each
equation by the necessary factor. Equation A now
becomes 10x 6y 10 Equation B
now becomes 9x 6y -48 Step 3 ADD the two
equations if using opposite signs (if not,
subtract) Step 4 Solve the equation. Step
5 Back-substitute this coordinate into any
equation to find the other coordinate. (Look for
easiest coefficients to work with.)
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Method 3 Elimination or Addition/Subtraction
Method(Goal Combine equations to cancel out
one variable.)
Step 1 Look for the LCM of the coefficients on
either x or y. (Opposite signs are recommended
to avoid errors.) Here -3y and 2y could be
turned into -6y and 6y Step 2 Multiply each
equation by the necessary factor. A becomes
10x 6y 10 B becomes 9x 6y -32 Step 3
ADD the two equations if using opposite signs (if
not, subtract) Step 4 Solve for the remaining
variable Step 5 Back-substitute this coordinate
into any equation to find the other coordinate.
(Look for easiest coefficients to work with.)

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Method 3 Elimination or Addition/Subtraction
Method(Goal Combine equations to cancel out
one variable.)
Step 1 Look for the LCM of the coefficients on
either x or y. (Opposite signs are recommended
to avoid errors.) Step 2 Multiply each
equation by the necessary factor. Step 3 ADD
the two equations if using opposite signs (if
not, subtract) Step 4 Solve for the remaining
variable Step 5 Back-substitute this coordinate
into any equation to find the other coordinate.
(Look for easiest coefficients to work with.)

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Example Elimination or Addition/Subtraction
Method(Goal Combine equations to cancel out
one variable.)
Step 1 Look for the LCM of the coefficients on
either x or y. (Opposite signs are recommended
to avoid errors.) Step 2 Multiply each
equation by the necessary factor. Step 3 ADD
the two equations if using opposite signs (if
not, subtract) Step 4 Solve for the remaining
variable Step 5 Back-substitute this coordinate
into any equation to find the other coordinate.
(Look for easiest coefficients to work with.)

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Example Elimination or Addition/Subtraction
Method(Goal Combine equations to cancel out
one variable.)
Step 1 Look for the LCM of the coefficients on
either x or y. (Opposite signs are recommended
to avoid errors.) Step 2 Multiply each
equation by the necessary factor. Step 3 ADD
the two equations if using opposite signs (if
not, subtract) Step 4 Solve for the remaining
variable Step 5 Back-substitute this coordinate
into any equation to find the other coordinate.
(Look for easiest coefficients to work with.)

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Method 3 Summary Elimination Methodor
Addition/Subtraction Method(Goal Combine
equations to cancel out one variable.)
Disadvantages Avoid this method if degrees
and/or formats of the equations do not match.
AdvantagesSimilar to getting an LCD, so this
is intuitive, and uses only integers until the
end of the problem.
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Three MethodsMethod 1 Graphing MethodMethod
2 Substitution MethodMethod 3 Elimination
Method or Addition/Subtractio
n Method
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Factoring

1. Factor GCF ? for any terms
2. Difference of Squares ? binomials
3. Sum or Difference of Cubes ? binomials
4. PST (Perfect Square Trinomial) ? trinomials
5. Reverse of FOIL ? trinomials
6. Factor by Grouping ? usually for 4 or more terms

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GCF

• 3x2-9x2
• 3x2(1-3)
• Take out what the two side share in common to
  simplify.

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GCF
3(1-6)26(1-6)2
3(1-6)2(13)
You can take out a glob and then combine with the
other globs.
Hint Remember that a glob can be part of your
GCF.
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Difference of Squares

• 25x2-4x2
• (5x-2)(5x2)

Recall these binomials are called conjugates.
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IMPORTANT!

• Remember that the difference of squares factors
  into conjugates . . .
• The SUM of squares is PRIME cannot be factored.
• a2 b2 ? PRIME
• a2 b2 ? (a b)(a b)

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Sum/Difference of Cubes

• x3-y3
• (x-y)(x2xyy2)

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Sum/Difference of Cubes

• x3 - y3
• (x - y) ( )
• Cube roots w/ original
• sign in the middle

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Sum/Difference of Cubes

• x3 - y3
• (x - y) (x2 y2)
• Squares of those cube roots.
• Note that squares will always be positive.

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Sum/Difference of Cubes

• x3 - y3
• (x - y) (x2 xy y2)
• The opposite of the product
• of the cube roots

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Sum/Difference of Cubes

• x3 - 8
• (x - 2)
• Cube roots of each Squares of those cube roots
  
• with same sign opp of product of roots in
  middle

(x2 4)
2x
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Special Case- 1ststep Diff of Squares
2nd step Sum/Diff of Cubes

• x6 64y6
• ( ) ( )
• ( )( ) ( )(
  )

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Special Case 1ststep Diff of Squares
2nd step Sum/Diff of Cubes

• x6 64y6
• (x3 8y3) (x3 8y3)
• ( )( ) ( )(
  )

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Special Case 1ststep Diff of Squares
2nd step Sum/Diff of Cubes

• x6 64y6
• (x3 8y3) (x3 8y3)
• (x2y)(x22xy4y2) (x2y)(x2-2xy4y2)

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PST

• x2-10x25
• (x-5)(x-5)
• (x-5)2
• Recall PST test
• If 1st 3rd terms are squares and the middle
  term is twice the product of their square roots.

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PST

• x2-2416
• (3x-4)(3x-4)
• Conjugates

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Reverse FOIL (Trial Error)

• 12x2-45x42
• (3x-6)(4x-7)
• Conjugates

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Reverse FOIL(Trial Error)

• Hint dont forget to read the signs
• ax2 bx c ? ( )( )
• ax2 bx c ? ( )( )
• ax2 bx c ? ( )( )
• positive
  product has larger value
• ax2 bx c ? ( )( )
• negative
  product has larger value

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Example 11 Factor by Grouping(4 or more terms)

• a(x-y)x-y(x-y)
• (x-y)3(a)

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Example 11 Factor by Grouping(4 or more terms)

• 8x-2y16x-4y
• 2(4x-y)4(4x-y)
• (4x-y)2(24)

2X2
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Factor by Grouping 3 X 1

• x2xyy2-4x2
• (x-y)(x-y)-4x2
• (x-y)-2x(x-y)-2x

• 3X1 (PST)

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Factor by Grouping 3 X 1
x2-1025-4x2 (x-5)(x-5)-4x2 (x-5)-2x(x-5)-2x
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Rational Expressions
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Rational Numbers

• Thinking back to when you were dealing with
  whole-number fractions, one of the first things
  you did was simplify them You "cancelled off"
  factors which were in common between the
  numerator and denominator. You could do this
  because dividing any number by itself gives you
  just "1", and you can ignore factors of "1".
• Using the same reasoning and methods, let's
  simplify some rational expressions.
• Simplify the following expression
•  
• To simplify a numerical fraction, I would cancel
  off any common numerical factors. For this
  rational expression (this polynomial fraction), I
  can similarly cancel off any common numerical or
  variable factors.
• The numerator factors as (2)(x) the denominator
  factors as (x)(x). Anything divided by itself is
  just "1", so I can cross out any factors common
  to both the numerator and the denominator.
  Considering the factors in this particular
  fraction, I get
• Then the simplified form of the expression is 
•  

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Additoin and Subtraction of Rational Numbers

•   3           5               8              
  (2)(4)            2                          
                                      20        
  20             20              (4)(5)           
  5
• Notice the steps we have done to solve this
  problem.  We first combined the numerators since
  the denominators are the same.  Then we factored
  both the numerator and denominator and finally we
  cross cancelled.

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Multiplication And Division of Rational Numbers

•  x2 - y2                          is a rational
  expression.(x - y)2  To simplify, we just
  factor and cancel
• (x - y)(x y)            x   y                 
                              (x - y)2
                   x - y

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• Quadratic Equations

97
Quadratic Equations

• A quadratic equation is an equation that can be
  written in this form.
• ax2bxc0
• The a,b, and c here represent real number
  coefficients. So this means we are talking about
  an equation that is a constant times the variable
  squared plus a constant times the variable plus a
  constant equals zero, where the coefficient a on
  the variable squared can't be zero, because if it
  were then it would be a linear equation.

98
Completing the Square

• a² 2ab b²(a b)².The technique is valid
  only when 1 is the coefficient of x².
•  1)  Transpose the constant term to the rightx²
  6x   -2
•  2)  Add a square number to both sides.  Add the
  square of half the coefficient of x.  In this
  case, add the square of 3x² 6x 9    -2 9.
• The left-hand side is now the perfect square of
   (x 3).
• (x 3)²    7.
• 3 is half of the coefficient 6.
• This equation has the form
• a²  b   which implies a   .        
   Therefore,x 3    
• x  -3 .
• That is, the solutions to
• x² 6x 2    0
• are the conjugate pair,
• -3 ,  -3 - .

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Quadratic Formula
100
Quadratic Formula

• on multiplying both c and a by 4a, thus making
  the denominators the same (Lesson 23),
• This is the quadratic formula.  

101
Discriminant

• The radicand  b² - 4ac  is called the
  discriminant.  If the discriminant is
• a)   PositiveThe roots are real and conjugate. 
• b) Negative  The roots are complex and
  conjugate. 
• c)   ZeroThe roots are rational and equal --
  i.e. a double root.

102
Factoring

• Problem 2.   Find the roots of each quadratic by
  factoring.
• a)  x² - 3x 2 b)  x² 7x 12
• (x - 1)(x - 2)  (x 3)(x 4)
• x 1  or  2.  x -3  or  -4.

103
In the next slides you will reviewFunctions D.
Quadratic functions explain everything we know
about how to graph a parabola
104
Functions

• A function is an operation on numbers of some set
  (domain) that gives (calculates) one number for
  every number from the domain. For example,
  function 3x is defined for all numbers and its
  result is a number multiplied by 3. The notation
  is yf(x). x is called the argument of the
  function, and y is the value of the function. The
  inverse function of f(x), is a different function
  of y that finds x that gives f(x) y.

105
Functions

• We have seen that a function is a special
  relation. In the same sense, real function is a
  special function. The special about real function
  is that its domain and range are subsets of real
  numbers R. In mathematics, we deal with
  functions all the time but with a difference.
  We drop the formal notation, which involves its
  name, specifications of domain and co-domain,
  direction of relation etc. Rather, we work with
  the rule alone. For example,
• f(x) x22x3
• This simplification is based on the fact that
  domain, co-domain and range are subsets of real
  numbers. In case, these sets have some specific
  intervals other than R itself, then we mention
  the same with a semicolon () or a comma(,) or
  with a combination of them
• f(x) \?(x1) 2-1xlt-2,x0
• Note that the interval xlt-2,x0 specifies a
  subset of real number and defines the domain of
  function. In general, co-domain of real function
  is R. In some cases, we specify domain, which
  involves exclusion of certain value(s), like
• f(x)
• 11-x,x?1
• This means that domain of the function is R-1 .
  Further, we use a variety of ways to denote a
  subset of real numbers for domain and range. Some
  of the examples are
• xgt1 denotes subset of real number greater than
  1.
• R-0,1 denotes subset of real number that
  excludes integers 0 and 1.
• 1ltxlt2 denotes subset of real number between 1
  and 2 excluding end points.
• (1,2 denotes subset of real number between 1
  and 2 excluding end point 1, but including
  end point 2.
• Further, we may emphasize the meaning of
  following inequalities of real numbers as the
  same will be used frequently for denoting
  important segment of real number line
• Positive number means x gt 0 (excludes 0).
• Negative number means x lt 0 (excludes 0).
• Non - negative number means x 0 (includes 0).
  
• Non positive number means x 0 (includes 0).

106
Quadratic Functions

• A quadratic function is one of the form f(x)
  ax2 bx c, where a, b, and c are numbers with
  a not equal to zero.
• 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.
• You know that two points determine a line. This
  means that if you are given any two points in the
  plane, then there is one and only one line that
  contains both points. A similar statement can be
  made about points and quadratic functions.
• Given three points in the plane that have
  different first coordinates and do not lie on a
  line, there is exactly one quadratic function f
  whose graph contains all three points. To graph
  simplify (see quadratic equations) and plot the
  points.

107
Simplifying expressions with exponents
108
Simplifying Exponents

• Use the Power of a Power Property, the Product of
  a Power Property, the Quotient of a Power
  Property, the Power of a Quotient Property, the

109
Simplifying exponents
Use These
110
Simplifying Exponents

• 32 315317

111
Simplifying expressions with radicals
112
Simplifying radicals

• When presented with a problem like , we
  dont have too much difficulty saying that the
  answer 2 (since 2x24 ). Our trouble usually
  occurs when we either cant easily see the answer
  or if the number under our radical sign is not a
  perfect square or a perfect cube. A problem
  like may look difficult because there
  are no two numbers that multiply together to give
  24. However, the problem can be simplified. So
  even though 24 is not a perfect square, it can be
  broken down into smaller pieces where one of
  those pieces might be perfect square. So now we
  have
• Simplifying a radical expression can also involve
  variables as well as numbers. Just as you were
  able to break down a number into its smaller
  pieces, you can do the same with variables. When
  the radical is a square root, you should try to
  have terms raised to an even power (2, 4, 6, 8,
  etc). When the radical is a cube root, you should
  try to have terms raised to a power of three (3,
  6, 9, 12, etc.). For example,
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

http//www.algebralab.org/lessons/lesson.aspx?file
Algebra_radical_simplify.xml
113
Simplifying Radicals

• Use the root of a power, power of a root, product
  of a root, and quotient of a root properties to
  solve.

114
In the next slides you will reviewMinimum of
four word problems of various types. You can mix
these in among the topics above or put them all
together in one section. (Think what types you
expect to see on your final exam.)
115
Suppose that it takes Janet 6 hours to paint her
room if she works alone and it takes Carol 4
hours to paint the same room if she works alone.
How long will it take them to paint the room if
they work together?
Click to see answer.

• First, we will let x be the amount of time it
  takes to paint the room (in hours) if the two
  work together.
•  
• Janet would need 6 hours if she did the entire
  job by herself, so her working rate is of
  the job in an hour. Likewise, Carols rate is
  of the job in an hour.
•  
• In x hours, Janet paints of the room
  and Carol paints of the room. Since the
  two females will be working together, we will add
  the two parts together. The sum equals one
  complete job and gives us the following equation
  
•  

116
Continued answer

• Multiply each term of the equation by the common
  denominator 12

Simplify
Collect like terms
Solve for x
117
Suppose Kirk has taken three tests and made 88,
90, and 84. Kirks teacher tells the class that
each test counts the same amount. Kirk wants to
know what he needs to make on the fourth test to
have an overall average of 90 so he can make an A
in the class.
Steps
118
Suppose a bank is offering its customers 3
interest on savings accounts. If a customer
deposits 1500 in the account, how much interest
does the customer earn in 5 years?

• I is the amount of interest the account earns.
• P is the principle or the amount of money that is
  originally put into an account.
• r is the interest rate and must ALWAYS be in a
  decimal form rather than a percent.
• t is the amount of time the money is in the
  account earning interest.
•  

If we want to find out the total amount in the
account, we would need to add the interest to the
original amount. In this case, there would be
1725 in the account
119
The Smiths have a rectangular pool that measure
12 feet by 20 feet. They are building a walkway
around it of uniform width.

• The length of the larger rectangle is
  , which simplifies to
•  
• Length larger rectangle
•  
• The width of the larger rectangle is
  , which simplifies to
•  
• Width larger rectangle
•  
• The area for the larger rectangle then becomes
•  
• Area larger rectangle
•  
• The pool itself has an area of
  square feet

120
Answer continued

• Rearrange the terms for easier multiplication and
  find the sum of 68 and 240.
• Multiply the binomials.
• Combine like terms and subtract 308 from each
  side.
• Factor.
• Solve each factor.
• Since dimensions of a pool and a walkway around a
  pool cannot be negativeour answer is that the
  width of the walkway is 1 foot.

121
Line of Best Fit or Regression Line
122
Line of best fit

• A line of best fit is a straight line that best
  represents the data on a scatter plot.  This
  line may pass through some of the points, none of
  the points, or all of the points.

123
Line of Best Fit

• You can find the line of best fit by estimation
  or by using graphing calculators.
• The line of best fit is good for estimating the
  average of the points on the graph.

124
Line of best fit (how to solve)

• 1. Separate the data into three groups of equal
  size according to the values of the
• horizontal coordinate.
• 2. Find the summary point for each group based on
  the median x-value and the
• median y-value.
• 3. Find the equation of the line (Line L) through
  the summary points of the outer
• groups.
• 4. Slide L one-third of the way to the middle
  summary point.
• a. Find the y-coordinate of the point on L with
  the same x-coordinate as the
• middle summary point.
• b. Find the vertical distance between the middle
  summary point and the line by
• subtracting y-values.
• c. Find the coordinates of the point P one-third
  of the way from the line L to the
• middle summary point.
• 5. Find the equation of the line through the
  point P that is parallel to line L.

125
Line of Best Fit

• Try to find the line of best fit for the points
  (3,9)(4,8)(6,6)(7,5)(9,7)(11,9)(13,12)(14,17)(13,1
  9).