Download P.1 Algebraic Expressions & Real Numbers

Chapter P
Prerequisites: Fundamental Concepts of
Algebra
P.1 Algebraic Expressions & Real
Numbers
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Objectives
Evaluate algebraic expressions
Use mathematical models
Find the intersection of 2 sets
Find the union of 2 sets
Recognize subsets of the real numbers
Use Inequality symbols
Evaluate absolute value
Use absolute value to express distance
Identify properties of the real numbers
Simplify algebraic expressions
Intersection of Sets
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What they have in common
A = {all tall children}
B = {all girls}
A intersect B = {all tall girls}
All children that are girls AND are tall
Union of Sets
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Combination of everything in both sets
A = {all tall children}
B = {all girls}
A union B = {all girls OR tall children} = {all girls
and all tall boys}
A group of biology majors are taking
Biology I & Chem. I. A group of chemistry
majors are taking Calculus, Chem. I and
Physics I. The Physics majors enrolled in
Calculus, Physics I, and Chem I. What is
the intersection of the 3 groups?
1.
2.
3.
4.
Students in biology, chemistry, & physics.
Students in chemistry.
Students in calculus.
Students in physics.
Absolute Value
• │x│ represents the distance between x and zero
• Distance is always a positive quantity, therefore
going left or right x units results in a distance of x
units
• │x - 2│ represents the distance between x and 2
• Distance is again always positive. (i.e. the
distance between 2 and 3 is 1 and the distance
between 2 and 1 is 1) │3 - 2│ = │1 - 2│ = 1
Real numbers are a field
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Commutative (addition & multiplication)
Associative (addition & multiplication)
Identity (additive = 0 & multiplicative = 1)
Inverse (additive = -x & multiplicative = 1/x)
Distributive (multiplication over addition)
• ALL these properties are useful when
manipulating algebraic expressions & equations
P.2
• Exponents and Scientific Notation
Objectives
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Use the product and quotient rules
Use the zero-exponential rule
Use the negative-exponent rule
Use the power rule
Find the power of a product
Find the power of a quotient
Simplify exponential expressions
Use scientific notation
Product & Quotient Rules
x  x  ( x  x  x)  ( x  x  x  x)  x
3
4
7
a
aaaaa
3 aa
3

a 
a
2
a
aa
aa
5
Example
• Simplify:
24ab c
83 a b b  c
3ab


2 5
2
5
16b c
8 2b c
2
3 5
3 5
2
5
24ab c
24
3
( 3 2 ) ( 5  5 )

a

b
c

ab
2 5
16b c
16
2
•
Quotient Rule explains
zero-exponent rule
y
Any real number
x
( y y)
0
x
x
divided by itself
y
x
(except 0) equals 1
• If x is any nonzero
number & y is an
exponent:
y
x

1
y
x
x 1
0
Quotient Rule Explains Negative
Exponent Rule
3
x
( 3 5 )
2
x
x
5
x
3
3
x
x
1
 3 2  2
5
x
x x
x
x
2
1
 2
x
Working with Negative Exponents
• In general, expressions are not considered
simplified when negative exponents are
present.
• A negative exponent in the denominator
becomes positive when moved to the
numerator
• A negative exponent in the numerator
becomes positive when moved to the
denominator
Raising an Exponent to an
Exponent (Power Rule)
• Exponents are multiplied
• WHY?
(a b )  (a b )( a b )( a b )  a b
2 3 3
(x )  x
m n
2 3
mn
2 3
2 3
6 9
When is your expression
simplified?
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No negative exponents are present
Each base appears only once
No parentheses remain
Example:
2
3 4
8 12
5 18
(3x y )
81x y
81x y


2 3
3 6
(2 xy )
8x y
8
Scientific Notation
• What is it? A number greater than or equal to 1 &
less than 10 (either pos. or neg.) multiplied by 10
raised to an exponent
• Example:
2.110
5
 5.13 10
7
Why Use Scientific Notation?
• It allows us to express very large numbers
or very small numbers in a more concise
manner.
• Diminishes the error in writing very large
or small numbers by eliminating the need
to have all zeros written. (easy to have
one too many or too few zeros)
Converting into Scientific Notation
345,000  3.45 100,000  3.45 10
5
691
6.9110
4
.000691 

 6.9110
6
1,000,000
10
2
Rules of Thumb
• Count decimal places you move to place
the decimal to the right of one non-zero
digit
• Large numbers are represented by
multiplying by ten raised to a positive
exponent
• Small numbers are represented by
multiplying by ten raised to a negative
exponent
P.3
• Radicals & Rational Exponents
Objectives
• Evaluate square roots
• Simplify (nth root of nth power)
• Use product & quotient rules to simplify square
roots
• Add & subtract square roots
• Rationalize denominators
• Evaluate & perform operations with higher roots
• Understand & use rational exponents
Principal Square Root
• It is true that 4 squared and (-4) squared both
equal 16, BUT the principal square root of 16 is 4
NOT -4
• By convention, the radical symbol represents the
positive (or PRINCIPAL) square roots of a number,
thus for real numbers, x, greater than or equal to 0:
x 0
Examples
250  25  10  5 10
1
1


9
3
36  64  6  8  2
Multiplying & Dividing with Radicals
(Roots)
• A product or quotient under a radical can
be written as the product or quotient of
separate radicals
• Products or quotients involving square
roots can be expressed as a single square
root involving products or quotients under
the radical
Adding & Subtracting Square Roots
• ONLY when you’re taking the square root of the
same number can you add or subtract square
roots
2 7 5 7  7 7
2 5
7
Simplify Expressions, then
Add/Subtract (if possible)
2 24  5 54  2 4  6  5 9  6
 2(2) 6  5(3) 6  4 6  15 6  11 6
What is a conjugate?
• Pairs of expressions that involve the sum
& the difference of two terms
• The conjugate of a+b is a-b
• Why are we interested in conjugates?
• When working with terms that involve
square roots, the radicals are eliminated
when multiplying conjugates
Multiplying conjugates which
involve square roots
( x  y)( x  y)
 x  x  xy  xy  y  y
 x  y  x y
2
2
Expressions with radicals in the
denominator are NOT simplified
• Eliminate the radical from the denominator by
multiplying by the numerator and the
denominator by the conjugate of the denominator
• Sometimes the result may not LOOK simpler!
6
3
2
6

3
3
2 (3 

3 (3 
18  6 3  3 2  6

93
18  6 3  3 2  6
6
3)
3)
Other Roots
• The nth root of a number means “what
number could you raise to the nth power to
get your original number?”
• You can take an odd root of a negative
number or a positive number.
• You can only take an even root of a
positive number.
Rules for other roots
• Add and subtract only same roots of same
number (i.e. you can add cube roots of 3
but NOT cube roots of 3 and cube roots of
4)
• Multiply & divide same roots following
same rules as square roots
Subtract, if possible, & simplify:
6 16  4 2
3
3
1) 2 14
3
2) 2 2
3
3) 4 2
3
4)8 2
3
Expressing roots as rational
exponents
• Any root can be expressed as a rational
exponent, then rules of exponents apply
n
xx
1
n
Expressions may involve
exponents AND roots
• If possible, it’s often easier to take the root first
(the rational exponent), then raise the value to
the other exponent
5
4
16  ( 16 )  (2)  32
4
5
5
P.4
• Polynomials
Objectives
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Understand the vocabulary of polynomials
Add & Subtract polynomials
Multiply polynomials
Use FOIL in polynomial multiplication
Use special products in polynomial
multiplication
• Perform operations with polynomials in
several variables
• A polynomial in x is many terms added or
subtracted with each term involving a
constant and x raised to a power.
• Only same powers of x can be
added/subtracted
• When multiplying polynomials, the
distributive property holds. (i.e. every term
in one polynomial must be multiplied by
every term in the other polynomial.
(3x  5)(3x  2 x  7)
3
3 x(3 x  2 x  7)  5(3 x  2 x  7)
3
3
 9 x  6 x  21x  15 x  10 x  35
4
2
3
 9 x  15 x  6 x  31x  35
4
3
2
Special Products
(a  b)( a  b)  a  b
2
2
(a  b)( a  b)  a  2ab  b
2
(a  b)( a  b)  a  2ab  b
2
2
2
(a  b)( a  b)( a  b)  a  3a b  3ab  b
3
2
2
(a  b)( a  b)( a  b)  a  3a b  3ab  b
3
2
2
3
3
P.5
• Factoring Polynomials
Objectives
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Factor out the greatest common factor
Factor by grouping
Factor trinomials
Factor difference of squares
Factor perfect square trinomials
Factor sum & difference of cubes
Use a general strategy for factoring
Factor expressions containing fractional &
negative exponents
Factoring strategies
• FIRST: Look for greatest common factor
• Group terms (if 4 or more) to find common
terms between groups
• If only 3 terms, rewrite into 4 terms by
multiplying leading coefficient by the
constant term (a times c), then rewrite bx
as the sum of 2 terms whose product of
their coefficients is ac (then group as in
previous item)
EXAMPLE
6 x  23 x  20
2
6  20  120
120  15  8
15 x  8 x  23 x, substitute
6 x  15 x  8 x  20
2
3 x( 2 x  5)  4( 2 x  5)
( 2 x  5)(3 x  4)
Factor by Recognition
• Difference of Squares
• Difference or Sum of Cubes
• Opposite signs cause all middle terms to cancel
out
x  y  ( x  y )( x  y )
2
2
x  y  ( x  y )( x  xy  y )
3
3
2
2
x  y  ( x  y )( x  xy  y )
3
3
2
2
Factor Completely
4 x  32 y
3
3
1)( 4 x  4 y )( x  8 y )
2
2
2)4( x  8 y )
3
3
3)4( x  2 y )( x  2 xy  4 y )
2
2
4)4( x  2 y )( x  2 xy  4 y )
2
2
P.6
• Rational Expressions
Objectives
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Specify domain of a rational expression
Simplify rational expressions
Multiply rational expressions
Divide rational expressions
Add & subtract rational expressions
Simplify complex rational expressions
Simplify fractional expressions that occur
in calculus
• Rationalize numerators
Domain restrictions
• No values can be substituted in for x that
would create a zero denominator or a
negative value under a positive root
Simplify rational expressions
• Factor numerator and denominator to
cancel common terms
• Do NOT forget that the terms cancelled
still were in the original expression,
therefore must be considered when stating
the domain
Adding & Subtracting Rational
Expressions
• Expressions MUST have a common
denominator to be added/subtracted
• Remember when creating a common
denominator, both the numerator &
denominator must be multiplied by the
same term, otherwise the resulting
expression will NOT be equivalent to the
original
Simplify
1
x
x  x2
2  x 1 2
x
1
x
x
x x2
x
 ( )
( )
x 2 x
x 1 2
x
x2 1
x2  2x
x2 1
x x2  2x



 
x(2  x)
x2
x(2  x) x
x2
x 2  1  x3  2 x 2
 x 3  3x 2  1


x(2  x)
x2  2x
Rational expressions that occur in
calculus
f ( x  h)  f ( x )
h
• To simplify this expression, you may have to
rationalize the numerator (if f(x) involves a root)
f ( x)  x  1, f ( x  h)  x  h  1
f ( x  h)  f ( x )
x  h 1  x 1

h
h
x  h 1  x 1 x  h 1  x 1
rationaliz e :

h
x  h 1  x 1
( x  h  1)  ( x  1)
h
1



h ( x  h  1  x  1) h ( x  h  1  x  1)
x  h 1  x 1
P.7 Equations
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Objectives
Solve linear equations in one variable
Solve linear equations containing fractions
Solve rational equations with variables in
the denominators
• Solve a formula for a variable
• Solve equations involving absolute value
Objectives continued
Solve quadratic equations by:
a) Factoring
b) Using the square root property
c) Completing the square
d) Using the quadratic formula
(WHEN TO USE WHICH METHOD?)
Use discriminant to determine # & type of
solutions
Solve application problems involving
quadratics
Solve radical equations
What is a linear equation in one
variable and what is “solving it”?
Only one variable (x or y, generally) is in the
equation and it is NOT squared or raised
to a power other than 1.
To “solve” the equation means to find the
value (or values) that would make the
equation true.
How do we solve an equation?
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Eliminate parentheses (distribute!)
Collect like terms (additive identity)
Isolate the variable (multiplicative identity)
Remember: it’s an EQUATION to start
with, meaning the left equals the right. It
will no longer be equal, if something is
done to one side and not the other!
• CHECK your solution in the original
equation: does it make it true?
EXAMPLE
• 4(2x-3) = 2(x+3)
• 1)Distribute to eliminate parentheses
8x-12 = 2x + 6
2)Collect x’s on one side & constants on
the other (use additive identity)
8x(-2x) – 12(+ 12) = 2x(-2x) + 6(+ 12)
6x = 18
3)Isolate the x (use multiplicative identity)
4) Check your solution in the original
4(2(3)-3) = 2(3+3)
4(3) = 2(6) YES!!
6 x 18

6
6
x3
Rational Equations
• Equations that involve fractions!
• The variable (x) could be in the numerator of the
denominator. IF the x is found in the
denominator, we must consider values x
canNOT take on. (i.e. zero denominator)
• EVEN after you’ve simplified an equation to
eliminate the fractions, you haven’t eliminated
the original restriction that may have been
present.
• With fractions, EITHER eliminate the fraction OR
get a common denominator (if denominators are
EQUAL, so are numerators)
Solve by getting a common
denominator
3
1
1
  , x  2,...WHY ??
x2 8
2
3(8)
1( x  2)
1( x  2)( 4)


( x  2)(8) 8( x  2)
2( x  2)( 4)
24  x  2
4x  8

8( x  2)
8( x  2)
 26  x  4 x  8
26  x  x  8  4 x  8  x  8
18  3 x, x  6
CHECK!!
Types of Equations
• Conditional: True under certain conditions
(could be one or several solutions)
• Inconsistent: Inconsistencies between the
2 sides (never true – NO solutions)
• Identity: One side of the equation is
identical to the other (doesn’t matter what
x is, infinitely many solutions)
Example
• Solve 3x – 6 = 3(x – 2)
Notice, after distributing on the
right, 3(x – 2) = 3x – 6
The left side is identical to the
right. No matter what values you plug in
for x, it will always be true.
The solution set is: {all reals}
THIS IS AN IDENTITY.
Example
• Solve: 4x – 8 = 4(x – 5)
• Distribute on right = 4x – 20
• Think: Can 4 times a number minus 8
possibly equal 4 times the same number
minus 20??? NO!!
• If you continue to solve, you get:
• 0x = -12 (Can 0 times a number ever equal -12?
NO!
• INCONSISTENCIES!! Solution: { }
Quadratic equations
Various methods to solve different
quadratics
What is a quadratic equation?
ax  bx  c  0
a, b, c  {Re als}
2
Zero-Product Rule
• If the product of two or more numbers is
zero, at least one of the numbers must
equal zero!
• If AB=0, then A=0 and/or B=0
– One or both of the terms must equal zero
Why is this important?
It allows us an easy way to solve an equation,
but FIRST make certain the expression is a
product that equals zero.
• A product involves FACTORS
• (2x-3)(x+2)=0
• 2x – 3 is a factor of the expression, as is
2+x
• Set each factor = 0
• 2x – 3 = 0, thus x = 3/2
• x + 2 = 0, thus x = -2
• SO, if EITHER x = 3/2 or x = -2, the
original expression = 0
• SO, solve by FACTORING if equation,
once equal to 0, is FACTORABLE
Often, you must get expression into
factored form FIRST:
Solve : 6 x  11x  4
2
6 x  11x  4  0
2
Factor : (3x  4)( 2 x  1)  0
3x  4  0
2x 1  0
4 1
x ,
3 2
Solving with square root property
• When would you use this approach?
– When one side of the equation is a perfect square
EXAMPLE:
(3  2 x) 2  5
(3  2 x) 2    5 (Why  ??)
3  2 x  5i (WHY ??)
 2 x  3  5i
 3  5i 3 5
x
  i
2
2 2
Solve by Completing the Square
• When can you use this method? ALWAYS
– However, if the expression is factorable or is
already a perfect square, those methods may
be more desirable
HOW does it work?
If you don’t have a perfect square, you create
one by adding a “well-chosen” zero (adding
the same term to both sides)
Decide what to add by determining what
additional term would create a perfect square
Solve : 2 x 2  4 x  7  0
EXAMPLE
7
2
2( x  2 x  )  0
2
7
2
x  2x  
2
( Note : ( x  1) 2  x 2  2 x  1)
7
5
2
x  2x 1   1  
2
2
5
2
( x  1)  
2
5
2
( x  1)   
2
5
5
x  1   i, x  1  i
2
2
Completing the square generalized
to any quadratic equation results in
the quadratic formula.
• When can you use it? ALWAYS. (However, it still
may be easier to factor & use zero-factor property
or take the square root if it’s already a perfect
square.)
2
ax  bx  c  0
 b  b  4ac
x
2a
2
Solve:
2x  4x  6
2
1.
2.
3.
4.
x = -1, 3
x = 1, -3
x = 2,3
x=2
What is the discriminant and why is
it useful to us?
• The discriminant is the part of the
quadratic equation that is under the
radical.
• Based on what is under that radical, we
can determine if our solution will be an
integer (is what’s under there a perfect
square?), an irrational (is what’s under
there a positive number that is NOT a
perfect square), or complex (is what’s
under there a negative number?)
What if your equation involves a
variable under a radical?
• In order to eliminate an nth root, you must
raise both sides of the equation to the nth
power.
• Be CERTAIN that you isolate the radical
(have it on one side of the equation by
itself) before you raise both sides to the
nth power.
What if the variable is found under
a radical twice in an equation?
• Isolate one radical and raise both sides to
the nth power.
• Then, isolate the other radical (it will not
have disappeared from the other side),
and raise both sides to the nth power
again.
What is x is raised to an exponent
that is NOT an integer?
• If the variable (or expression involving a
variable) is raised to the (m/n) exponent, you
must isolate that expression and then raise
BOTH sides to the (n/m) power.
• WHY?? When you raise one exponent to
another, you multiply the 2 exponents.
( a  b)
m
n
m
n
n
m
(( a  b) )  a  b
What if the equation involves an
expression inside absolute value
brackets?
• Recall what absolute value means: What is
within those brackets could be positive or
negative and still have the same overall value.
3 x  7  15
I )3 x  7  15
22
3 x  22, x 
3
II )  (3 x  7)  15
 3 x  7  15
8
 3 x  8, x 
3
Solve:
3x  7  2  1
1.
2.
3.
4.
No solution.
{7/3}
{10/3, 4/3}
{-7/3, 7/3}
P.8
• Modeling with Equations
Objectives
• Use equations to solve problems
Solving Word Problems
• 1) Carefully read the problem
• 2) Determine what do you know and
what do you want to know
• 3) Identify variables
• 4) Develop equation relating what you
know & what you want to know
• 5) Solve the equation & check (correct?)
• 6) Make certain you answered the
question you were being asked!
EXAMPLE
• You need to drive from Chicago to your
cousin’s house in Omaha (a distance of
550 miles) at an average 65 mph on the
Interstate highway. What time should you
leave if you have to be at your cousin’s at
3:30 pm?
• What do you want to know?
– How long will it take you to drive? (x = time)
– What time must you leave?
• What do you know?
– Total distance you’ll travel (550 miles)
– Speed (65 miles per hour)
• What is the relationship between known &
unknown?
– Distance = Rate x Time
– 550 miles = 65 mph x (X) (cont. on next slide)
550mi
x
 8.5hr
mi
65
hr
Did you answer the question?
NO – WHEN should you leave?
In order to arrive at 3:30pm, you
leave 8.5 hrs earlier, which would
be at 8:00 am.
EXAMPLE
• You have been asked to make an
aluminum can (cylindrical shape) to hold
300 ml of your product. The can is to be
10 cm high. How much aluminum (in
square cm) do you need?
• What do you know?
– Can holds 300 ml (the volume!)
– Height = 10 cm
What do you want to know?
-How much material you will need (surface area).
What relates the known & unknown?
For cylinders:
V  r h
2
SA  2r  2rh
2
(example continued)
V  300ml  300cm  r (10cm)
3
3
300cm
2
r 
 9.55cm
 10cm
r  3.1cm
2
2
Now find surface area!
(answer the question!)
(remember, a cylinder is just
2 circles and a rectangle)
SA  2r  2rh
2
SA  2 (3.1cm)  2 (3.1cm)(10cm)
2
SA  255cm
2
Ava purchased a new ski jacket, on
sale for $66.50. The coat had been
advertised as 30% off! What was
the original cost?
1.
2.
3.
4.
$95
$86.50
$90
$96.50
P.9
• Linear Inequalities and Absolute Value
Inequalities
Objectives
•
•
•
•
Use interval notation.
Find intersections & unions of intervals.
Solve linear inequalities.
Recognize inequalities with no solution or
all numbers as solutions.
• Solve compound inequalities.
• Solve absolute value inequalities.
Linear inequalities
• For equalities, you are finding specific
values that will make your expression
EQUAL something. For inequalities, you
are looking for values that will make your
expression LESS THAN (or equal to), or
MORE THAN (or equal to) something.
• In general, your solution set will involve an
interval of values that will make the
equation true, not just specific points.
What if you have more than one
inequality?
• If two inequalities are joined by the word
“AND”, you are looking for values that will
make BOTH true at the same time. (the
INTERSECTION of the 2 sets)
• If two inequalities are joined by the word
“OR”, you are looking for values that will
make one inequality OR the other true (not
necessarily both), therefore it is the
UNION of the 2 sets.
What IS an absolute value
inequality?
• Recall that absolute value refers to the
expression inside the brackets being either
positive or negative, therefore the absolute value
inequality involves 2 separate inequalities
• IF absolute value expression is LESS THAN a
value, you’re looking for values that are WITHIN
that distance (intersection of the 2 inequalities)
• IF absolute value expression is MORE THAN a
value, you’re looking for values that are getting
further away in both directions (union of the 2
inequalities)
If the absolute value is greater than a
number, you’re considering getting further
away in both directions, therefore an OR.
(get further away left OR right)
• See next slide for example:
2x  4  3
2x  4  3
7
x
2
OR  ( 2 x  4)  3
 2x  8  3
5
x
2
5
7
x   x 
2
2
• If, however, the absolute value was LESS
than a number (think of this as a distance
problem), you’re getting closer to your
value and staying WITHIN a certain range.
Therefore, this is an intersection problem
(AND)
• Same problem as before, but solved as a
LESS than inequality. (next slide)
2x  4  3
2x  4  3
7
x
2
AND  (2 x  4)  3
 2x  8  3
5
x
2
5
7 5 7
 x   x  , , 
2
2  2 2
Don’t leave common sense at the
door!
• Remember to use logic!
• Can an absolute value ever be less than
or equal to a negative value?? NO!
(therefore if such an inequality were
presented, the solution would be the
empty set)
• Can an absolute value ever be more than
or equal to a negative value?? YES!
ALWAYS! (therefore if such an inequality
were given, the solution would be all reals)