Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Title: Mathmatics 1 Lecture notes
Description: These are the intro to Mathmatics 5/6 college notes
Description: These are the intro to Mathmatics 5/6 college notes
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
MATH 21
College Algebra 1
Lecture Notes
MATH 21
3
...
(a + b)2 = a2 + 2ab + b2
...
(a − b)2 = a2 − 2ab + b2
...
(a + b)(a − b) = a2 − b2
...
(a + b)(x + y) = ax + ay + bx + by
...
a3 + b3 = (a + b)(a2 − ab + b2 )
...
a3 − b3 = (a − b)(a2 + ab + b2 )
...
1
...
4x + 4
3
...
36a2 − 12a + 1
7
...
x2 − (y + 2)2
4
...
x5 − x
5
...
3xy + 15x − 2y − 10
Page 1 of 1
College Algebra 1
MATH 21
4
...
You can’t divide by 0
...
−a = −b = − a
...
−a = a
...
3
...
x2 − 9
x2 + 3x
a3 + 7a2 + 10a
a3 − 7a2 − 18a
8
...
6
...
3n2 + 16n − 12
7n2 + 44n + 12
9
...
12
1
...
12y
Page 1 of 1
College Algebra 1
4
...
b
MATH 21
4
...
a · d = a·c = ac
...
5
...
9n2 −12n+4
n2 −4n−32
x2 −3xy+2y 2
x2 y−xy 2
2 4
·
3 5
÷
x2 −3x−18
x−1
12 4
·
9 18
x2 y 2
3
...
2 4
÷
3 5
9
...
6
...
Perform the indicated operations and simplify
...
x+2
x−3
·
÷
3n3 −2n2
n2 +4n
x2 −6x+9
x2 −4
5a2 +20a
a3 −2a2
·
÷
x−2y
x3 −xy 2
a2 −a−12
a2 −16
10
...
bc
MATH 21
4
...
a + c = a+c
...
a − c = a−c
...
Don’t “cancel” anything that is not being multiplied
...
6
...
− +
2 6 5
7
...
2
3
8
...
4
4+x
Perform the indicated operations and simplify
...
7 7
2
...
3
7
+
8 10
Page 1 of 2
−
2
2+x
MATH 21
4
...
4
2x−3
−
5x
3−2x
14
...
3
2x+1
+
2
2x+4
15
...
3x
−2
x−4
16
...
2x+12
x+2
17
...
4 Rational Expressions and Complex Fractions
Steps for Simplifying Fractions
Step 1
...
Step 2
...
Step 3
...
Step 4
...
Step 5
...
Step 6
...
9
x2 +2x+1
4a−4
a2 −4
−
5
x2 −1
6
b2 −3b−54
6
...
4
...
Perform the indicated operations and simplify
...
x23x + x
−6x
3
x+1
+
−
10
b2 +5b−6
3
a+2
3
...
t−3
2t+1
+
2t2 +19t−46
2t2 −9t−5
−
t+4
t−5
Page 1 of 2
+
x+5
x2 −1
−
3
x−1
College Algebra 1
MATH 21
4
...
Technique 1
...
Technique 2
...
2
3
− x+3
x−3
2
5
− x−3
x2 −9
8
...
9
...
1 −
10
...
Page 2 of 2
3
n−5
1−
1
a
a
+4
−2
4
n−5
MATH 21
4
...
Find the LCD
...
Multiply both sides by the LCD
...
Gather like terms
...
If necessary, get 0 on one side of the equation and factor the other side
...
Step 5
...
Solve each equation
...
x+2 + x−1 = 3
5
6
5
2
...
n −
3
n
=
26
3
11
3n
6
...
x+2
5
+
3
2x−1
x−1
6
=
=
=
1
x
7
...
x+6
27
x
x−4
−
2
x+3
3
5
5
3x+2
Page 1 of 2
=
20
x2 −x−12
MATH 21
4
...
A proportion is a statement that two ratios are equal
...
If a = d , then ad = bc
...
The sum of a number and its reciprocal
25
is 12
...
9
...
The ratio of
men to women is 5 to 7
...
The sum of two numbers is 84
...
What are the two numbers?
11
...
When
the larger number is divided by the smaller
number, the quotient is 2 and the remainder is 4
...
)
Divisor
Divisor
12
...
How much money
will the son receive?
13
...
If the ratio of its width to its
length is 2 to 5, then what are its dimensions?
Page 2 of 2
MATH 21
4
...
Find the LCD
...
Multiply both sides by the LCD
...
Gather like terms
...
If necessary, get 0 on one side of the equation and factor the other side
...
Step 5
...
2
...
1 +
+
2
n−6
4
...
4
1
...
5y−4
6y 2 +y−12
6
...
7 More Fractional Equations and Applications
College Algebra 1
Steps for Solving Word Problems
Step 1
...
Step 2
...
Step 3
...
Step 4
...
Step 5
...
Distance = Rate · T ime
7
...
Solve
x+3
z−2
=
1
B
=
+
2
y
1
C
11
...
My son
(Andrew) can clutter up the living room
in 40 minutes
...
for x
...
Harry flies his broom 96 miles in the
same time it takes Luna to fly her thestral
56 miles
...
12
...
If
it takes them 1 hour to clean a sea cucumber when working together, then how long
will it take Nemo working alone?
10
...
Duke
can mow the lawn in 2 hours
...
It takes Lester 1 hour longer than it
takes Chester to paint a picture
...
1 Using Integers as Exponents
Properties of Exponents
1
...
(bm )n = bmn
3
...
bm = bn−m
6
...
1
...
(3−2 )−3
10
...
(−4x−1 y 2 )(6x3 y −4 )
3
...
4
...
108a−5 b−4
9a−2 b
13
...
x−1 · x3
14
...
(x4 )5
8
...
a−2 + a−1 b−2
9
...
a =
b
7
...
2 Roots and Radicals
College Algebra 1
Radicals and nth Roots
√
A (non-negative) number a is an nth root of b if and only if an = b
...
Simplify each of the following expressions
...
2 16
2
...
√
3
125
Properties of Radicals
√
√
If b, c ≥ 0,
1
...
Write each expression in simplest radical
form
...
18
6
...
√ √
n
anb
144
36
9
...
3
...
3 3 24
Rules for Simplifying Radicals
1
...
2
...
3
...
√
√5
18
√
3√2
4 5
15
...
12
...
14
...
√
√4
3
Page 1 of 1
3
...
3 Combining and Simplifying Radicals
College Algebra 1
Adding Radical Expressions
When we add radicals we treat the simplified radical much like a variable
...
√
√
1
...
6 12 + 3 − 2 48
4
...
3 18 − 8 2
√
6−
√
12
Rules for Simplifying Radicals
1
...
2
...
3
...
6
...
√
3
24a8 b9
8
...
Write each expression in simplest radical
form
...
50y
√
√ 5y
18x3
10
...
4 8n + 3 18n − 2 72n
√
√
√
12
...
4 Products and Quotients Involving Radicals
Multiplying Radicals
√
√ √
Recall that n bc = n b n c
...
√
√
1
...
√
√
√
2x( 12xy − 8y)
√
√
2
...
( 7 − 2)( 7 − 8)
√
√
3
...
( 3 14)( 3 12)
5
...
−2 3(3 12 − 9 8)
√
√ √
√
9
...
3 3 3(4 3 9 + 5 3 7)
√
√ √
√
11
...
4 Products and Quotients Involving Radicals
College Algebra 1
Conjugates and Dividing Radicals
A pair of expressions a + b and a − b are said to be conjugate
...
√
12
...
√
2 7− 2
Simplify each expression
...
( 7 + 3)( 7 − 3)
√
17
...
√
x
x−1
5
10 − 3
√
√
2+ 3
√
18
...
√
3 2−5
19
...
5 Equations Involving Radicals
College Algebra 1
Solving Radical Equations
If a = b, then an = bn
...
Solve each equation and check your solution
...
4x = 6
5
...
4 x = 3
6
...
x + 1 + 5 = 3
7
...
√
2y − 3 = 5
√
√
√
6x + 5 =
7x − 6 −
√
√
2x + 10
5x + 2 = 0
x2 + 3 − 2 = 0
8
...
MATH 21
9
...
√
3
3x − 1 =
College Algebra 1
√
3
2 − 5x
14
...
10
...
5 Equations Involving Radicals
√
√
√
n−3+ n+5=2 n
√
11
...
√
−x − 6 = x
Page 2 of 2
MATH 21
5
...
b n = n b
College Algebra 1
m
2
...
1
1
...
1
2
2x 3 y 5
√
√
n m
b = ( n b)m
3
1
2
...
(−27) 3
1
12
...
−27 3
12x 3
1
8x 2
4
5
...
4− 2
7
...
( 2)( 4 2)
2
3
8
...
5
(2m + 3n) 7
9
...
√
4
√
3
14
...
2
1
10
...
√
4
4
Page 1 of 1
leave
in
radical
form
...
1 Complex Numbers
College Algebra 1
Complex Numbers
We define i to be a number such that i2 = −1
...
We
call a the real part and b the imaginary part
...
T
F
7 is a complex number
...
T
F
The real part of 3i is 0
...
That is, we treat i as we would a variable
...
3
...
(6 − 3i) − (4 − 4i)
Multiplying Complex Numbers
To multiply two complex numbers we treat i as we would a variable
...
Note that the rule ab = a b only works when a
and b are non-negative numbers
...
6
...
(3 + 2i)(4 − 3i)
Dividing Complex Numbers
To simplify a fraction of complex numbers, we multiply the denominator by its conjugate
...
Divide or simplify as indicated
...
4 − 3i
8
...
3+i
i
10
...
2 Quadratic Equations
College Algebra 1
Quadratic Equations
A quadratic equation is an equation containing one variable with highest exponent 2
...
Remember, that if ab = 0, then either a or b must be zero
...
1
...
24x2 + x − 10 = 0
2
...
√
x=x−2
A Convenient Fact
√
For any real number a, the equation x2 = a is true if and only if x = ± a
...
5
...
(x − 3)2 = 36
6
...
(4y + 5)2 = 18
Right Triangles
A right triangle with sides of length a and b and hypotenuse of length c satisfies the
Pythagorean Theorem: a2 + b2 = c2
...
9
...
A rectangular room is 8 feet by 15 feet
...
4
6
Page 1 of 1
MATH 21
6
...
An equation of
√
the form (x − b)2 = c has solution x = b ± c
...
Fill in the blank with the number that makes
the trinomial a perfect square
...
x2 + 10x +
2
...
x2 + 7x +
Completing the Square
We can solve a quadratic equation x2 + bx + c = 0 by subtracting c to the right-hand side
and then completing the square
...
If the coefficient of x2 is not
1, then we must first divide both sides of the equation by that coefficient
...
4
...
y 2 − 6y = −10
5
...
2x2 + 4x = 6
6
...
4x2 − 8x = −3
Page 1 of 1
MATH 21
6
...
2a
Solve each equation
...
x2 − 3x − 54 = 0
...
3x2 − 2x + 5 = 0
2
...
22t2 + 11t − 33 = 0
The Discriminant and a Useful Check
The part under the radical b2 − 4ac is called the discriminant
...
If b2 − 4ac = 0, there is one real solution
...
Checking Sum of Roots: The two solutions of ax2 + bx + c = 0 will always add up to
b
−a
...
5
...
2a2 − 6a + 1 = 0
Page 1 of 1
MATH 21
6
...
I suggest
1
...
Use the quadratic formula
...
12
t
18
t+8
9
2
Solve each equation
...
2x2 + 3x − 4 = 0
4
...
(x + 3)(2x + 1) = −3
5
...
2
x
+
5
x+2
=1
−
=
6
...
5 More Quadratic Equations and Applications
7
...
What is the number?
College Algebra 1
10
...
The length of the hypotenuse is 8 feet more
than the remaining side
...
An 8-inch by 10-inch picture is surrounded by a frame of uniform width
...
Find the width of the
frame
...
Bart’s time to travel 20 miles on his
skateboard is 1 hour less than the same as
Lisa’s time to travel 42 miles on her bike
...
Ike and Mike are brothers that have
square bedrooms
...
The total area of their
rooms is 221 square feet
...
The perimeter of a rectangle is 52
inches
...
What are the dimensions of
the rectangle?
Page 2 of 2
MATH 21
2
...
6 Inequalities and Interval Notation
College Algebra 1
Intervals
To solve an inequality, we isolate the variable just like solving an equation
...
Solve each inequality
...
1
...
−x + 6 ≥ 11
4
...
x − 1 < 2x − 2
5
5
(x + 1) ≤ −
4
2
Compound Inequalities
We use the words “and” and “or” in mathematics to form compound statements
...
The union of A and B (denoted A ∪ B) is the set of all elements in either
A or B or both
...
Write the solution set
in interval form and graph its solution set on
a number line
...
x > 1 and x < 4
6
...
18 ≤ −2x + 4 ≤ 10
8
...
What must he bowl in the third
game to have an average of at least 160 for
the three games?
Page 1 of 1
MATH 21
6
...
Solve each inequality and graph its solution
set on a number line
...
(x − 3)(x − 2) > 0
4
...
9x2 − 6x + 1 ≤ 0
2
...
2x3 + 10x2 > 0
3
...
6 Quadratic and Other Nonlinear Inequalities
College Algebra 1
Rational Expressions and Inequalities
f (x)
To solve an inequality of the form
> 0, we break up the number line into test
g(x)
intervals whose endpoints are the roots of f (x) and g(x)
...
8
...
10
...
Solve each inequality and graph the solution
set on a number line
...
x−2
x+3
≥1
x−4
x
≤0
x−1
4−x
≥0
x
Page 2 of 2
MATH 21
8
...
The graph of a
parabola resembles a valley
or an arch
...
4
...
2
...
6
...
1
...
y = − 1 x2
2
y = x2 − 4
y = x2 + 2
y = (x − 4)2
Page 1 of 2
MATH 21
8
...
Horizontal Shift by h units —
y = (x − h)2
2
...
Vertical Stretch/Reflect by a factor of a —
y = ax2
A parabola with vertex at the point (h, k) that is a times as ‘tall’ as y = x2 has equation
y = a(x − h)2 + k
...
9
...
11
...
y = −
...
Write the equation of the parabola pictured here
...
How does the graph of y = 4x2 compare to the graph of y = 2x2 ?
1
15
...
2 More Parabolas and Some Circles
College Algebra 1
Parabolas
The general approach to graphing a parabola of the form y = ax2 + bx + c is to convert it
to the form y = a(x − h)2 + k by completing the square
...
2
...
y = −x2 − 2x + 3
5
...
1
...
The standard
form of the equation of a circle of radius r centered at (h, k) is
(x − h)2 + (y − k)2 = r2
...
Plot the circle: (x − 4)2 + (y − 5)2 = 16
7
...
2 More Parabolas and Some Circles
Graph each circle
...
x2 + y 2 + 8x − 6y = −16
College Algebra 1
Write the equation of each circle
...
12
...
Center at (1, 6) and r =
9
...
Center at (−2, 3) and r = 1
10
...
Find the equation of the circle that
passes through the origin and has its center
at (3, 4)
...
Find the equation of the circle that
passes through the origin and has its center
at (−5, 12)
...
x2 + y 2 = −1
Page 2 of 2
MATH 21
8
...
The longer line segment
is called the major axis, and the shorter is called the minor axis
...
1
...
x2 + 4y 2 = 16
3
...
9x2 −36x+4y 2 −24y+36 = 0
5
...
How are the graphs of x2 + 4y 2 = 36 and
4x2 + y 2 = 36 related?
Page 1 of 1
MATH 21
8
...
Hyperbolas are characterized by two
symmetric parts that are bounded by a pair of asymptotes
...
5
...
4y 2 − 25x2 = 64
7
...
(y − 2)2 − (x − 5)2 = 4
1
...
4x2 − 9y 2 = 16
3
...
4
...
4 Graphing Hyperbolas
College Algebra 1
9
...
xy = 1
11
...
8
...
12
...
y 2 + y + 3x2 = 7
14
...
4x2 + 4y 2 = 16
16
...
−7x2 + 47x + 2y 2 + y = 800
Page 2 of 2
MATH 21
10
...
When we solve a system
of equations, we find all ordered pairs that satisfy each of the equations in the system
...
Solve the systems of equations by graphing
...
x − y = −1
2
...
Solve one equation for one variable in terms of the other variable
...
Substitute in place of that variable in the other equation
...
Solve that equation
...
Plug that solution into the other equation to get the value of the other variable
...
x − y = −4
3
...
9x − 2y = −38
5x + y = 0
5
...
−x + 4y = −22
−4x + 7y = −41
Page 1 of 2
MATH 21
10
...
2x + 3y = 7
7
...
9
...
The perimeter of a rectangle is 32
inches
...
find the dimensions of the rectangle
...
Suppose that Gus invested $8,000, part
of it at 8% and the remainder at 9%
...
How much did he invest at each rate?
12
...
Goofus has $350 in his bank
account
...
How many weeks will it take for
Gallant’s savings to catch up with Goofus’s
savings?
Page 2 of 2
MATH 21
10
...
Choose one variable to eliminate
...
Multiply both equations by suitable numbers so that the coefficients of that variable
are negatives of each other
...
Add the two equations to get a new equation
...
Solve the new equation
...
Plug that solution into the other equation to get the value of the other variable
...
x − y = −4
1
...
3x + y = 4
2x + 2y = 4
3
...
2x − 4y = 10
−4x + 8y = −20
6
...
It costs $130 to
buy 3 football tickets and 2 basketball tickets
...
At a certain store Ding Dongs cost $3
per box, and Twinkies cost $4 per box
...
1
x
2
+ 1y = 5
3
5x − 2y = 18
Page 1 of 2
MATH 21
10
...
4x − 3y = 1
8
...
College Algebra 1
12
...
What are the dimmensions of a quidditch field?
8x + 7y = 4
6x − 3y = 3
13
...
The pouch contains 29 coins for a
total of $1
...
How many dimes are in the
pouch?
10
...
The units digit of my age is one more
than three times the tens digit
...
How old am I?
11
Title: Mathmatics 1 Lecture notes
Description: These are the intro to Mathmatics 5/6 college notes
Description: These are the intro to Mathmatics 5/6 college notes