Algebra Notes

1.1 Real Numbers and Operations

Sets of real numbers:
-natural numbers (counting numbers): {1, 2, 3, 4, 5...}
-whole numbers (counting numbers and 0): {0, 1, 2, 3, 4, 5...}
-integers (whole numbers and their opposites): {...-3, -2, -1, 0, 1, 2, 3...}
-rational numbers (any number which can be written in the form of a/b where a and b are integers and b is not 0)
-irrational numbers (numbers with not-rational, non-terminating and non-repeating decimals): {pi}
-real numbers (the combination of rational and irrational numbers)

Absolute value is the distance of a number from zero. For instance, |-14| = 4.

Additive inverse: if a+ b = 0 then a and b are additive inverse.

1.2 Multiplication and Division of Real Numbers

positive same signs:
(positive)(positive) = (positive)
positive/positive = positive

different signs:
(positive)(negative) = negative
positive/negative = negative
negative/positive = negative

negative same signs:
(negative)(negative) = positive
negative/negative = positive

-property of division: when you divide a number, it is equivalent to multiplying by its reciprocal.

1.3 Algebraic Expression

Algebraic expression consists of numerals, variables or other mathematical symbols.
3x, -4, x + 5y

Evaluate the expression (plug in the variables for the unknowns):

3x^2+2x-5 for x = -2



1.4 The Distributive Property

a (b + c) = ab + bc

Factoring is the opposite process to this property.

6x + 15 = 3(3x = 5)

1.5 Solving Equations

if a = b then a + c = b + c


a + 4 = 18
(subtract 4 from each side)
a = 14

1.6 Problem Solving: Writing Equations

Steps for solving word problems:
I. Define variables for anything unknown in the problem.

II. Write any equations needed to solve the problem.

III. Show all steps taken to get to the solution.

IV. Write the solution in a complete sentence, with the proper units/labels.

-each step always uses Roman numerals to show order of steps.

1.7 Exponential Notation

81 = Standard form
81 = 8.1 x 10^1
81 = 9 x 9
81 = 3 x 3 x 3 x 3
81 = 3^4

3^4 is the exponential form of 81.

-exponent means reciprocal
b^-n = 1/b^n

in b^-n, b is the base and n is the exponent.

1.8 Properties of Exponents

(a^m)(a^n) = a^(m+n)

a^m/a^n = a^(m-n)

(a^m)^n = a^mn

= (a^m)^p (b^n)^p
=a^(mp) b^(np)

(a^m/b^n)^p = a^(mp)/b^(np)

1.9 Scientific Notation

Scientific notation is a method to express very large numbers. This method is called scientific notation. Scientific notation is based on powers of the base number 10.

The number 123,000,000,000 in scientific notation is written as :
1.23 × 10^11

The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called the base. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 × 10^11 the number 11 is referred to as the exponent power of ten.

To write a number in scientific notation:

Put the decimal after the first digit and drop the zeros.

In the number 123,000,000,000 the coefficient will be 1.23

To find the exponent count the number of places from the decimal to the end of the number.

In 123,000,000,000 there are 11 places. Therefore we write 123,000,000,000 as:
1.23 × 10^11

Exponents are often expressed using other notations. The number 123,000,000,000 can also be written as:
1.23E+11 or as 1.23 × 10^11

For extremely small numbers we use a similar approach. Numbers less than 1 will have a negative exponent
a millionth of a second is:
0.000001 sec. or 1.0E-6 or 1.0^-6

2.1 Solving More Equations

Zero Product property: For any real numbers, ab = 0, only if a = 0 or b = 0.


4x = 0
4x/4 = 0/4
x = 0

2.3 Solving Formulas

a = 1/2bh for b

(multiply both sides by 2)

2a = bh

(divide both sides by h)

2a/h = b

2.4 Solving Inequalities

Inequalities are equations that use the signs:

< (less than)
> (greater than)
<= (less than or equal to)
>= (greater than or equal to)
/= (is not equal to)

These equations are solved just the same as you would solve any other equation accept for if you divide both sides by a number than you will have to flip the symbol in the opposite direction.

If graphing an inequality on a line, a shaded in circle around a point represents the sign "less than or equal to" or "greater than or equal to". If the circle is not shaded in, than it represents the sign "less than" or "greater than".

2.5 Problem Solving Using Inequalities

Quiz grades are 73, 75, 89, and 91.
What is the lowest grade needed to obtain an average of 85

I. x = next quiz grade

II. (73 + 75 + 89 + 91 + x)/5 >= 85

III. (73 + 75 + 89 + 91 + x)/5 >= 85
(328 + x)/5 >= 85
(multiply both sides by 5)
328 + x >= 425
(subtract 328 from both sides)
x >= 97

IV. The lowest grade that is needed to obtain an average of 85 is 97.

2.6 Compound Inequalities

Solve and Graph:

x + 5 > 2 and x - 4 < 6
x > -3 and x < 10
(graph by making a line between -3 and 10, do not forget to make the circles on each point hollow)

2x + 5 > 8 or -2 -7 >= 11
2x > 3 or -2x -7 >= 18
x > 3/2 or x <= -9
(graph by making rays pointing in opposite directions form each number in the solution, do not forget to make the circles on each point hollow or shaded appropriately.)

2.7 Absolute Value


|3x| = |3||x|
= 3|x|

|x^2| = x^2
(the absolute value is not needed when the variable has an even power)

Inequalities with absolute value:

1st drop the absolute value
2nd drop the absolute value, switch the inequality and do the opposite of the original number.

If the original inequality is >=, > use "or".
If the original inequality is <=, < use "and".

3.1 Relations and Ordered Pairs

Cartesian coordinate plane: a two dimensional number line to locate specific points.

(x, y) where x is horizontal and y is vertical.

The set first coordinate in a relation is called the domain. The set of second coordinates in a relation is the range.


{(a, 1), (c, 8), (v, -13), (w, 0)}

domains: {a, c, v, w}
ranges: {1, 8, -13, 0}

3.2 Graphs

The points (x, y) are graphed on a coordinate plane where x is the horizontal axis and y is the vertical axis.

Equations such as y = x^2 + 3 will result in a parabolic shaped graph.

3.3 Functions

A function is a relation in which every member of the domain is paired with at most one member in the range. The x coordinates do not repeat.

Is the following a function?

{(4, 8), (9, -2)}
Yes, because the x-coordinates do not repeat.

If a relation is a function, the equation can be written in the form of f(x).
(Reads "f of x". The f(x) = y.

3.6 More with Equations of Lines

point-slope form: y - y = m ( x - x ) where m = slope

slope-intercept form: y = mx+b where m = slope and y = y-intercept = (0, b).

3.7 Parallel and Perpendicular Lines

Parallel lines:
-never intersect
-have different y intercepts
-have the same distance apart at any point

Perpendicular lines:
-have slopes that are opposite reciprocals of each other (m#1 times m#2 = -1, so m#1 = -1/m#2)
-form four right angles
-intersect only once

3.9 More on Functions

Step functions are also called greatest integer functions. y = |[x]|, that means that the greatest integer is not greater than x.

4.1 Systems of Equations with Variables

Systems of equations are equations with two or more variables. For every variable there should be at least two equations.

Solutions to a system:

-graphically is where the lines intersect.
- 1 solution
-no solution (parallel)
-infinite solutions (same lines)

4.2 Solving Systems: by Substitution Method

1. Solve for any variable in either equation.
2. Substitute the expression into the other equation for the solved variable.
3. Solve for the remaining variable.
4. Substitute the value found into any equation with both variables.
5. Solve to find the other variable.
6. Write the answer as a coordinate point.

4.3 Solving Systems: by Linear Combination Elimination


(the "2y" and the "-2y" cancel each other out. Combine the rest.)
(divide by 8)
(substitute "4" into any equation for "x" and solve)
(subtract the "12")
(divide by "-2")
(write the answer as a point in the (x, y) format)
(4, 5)

4.6 Consistent and Dependent Systems

consistent system - a system of equation with at least one solution

inconsistent system - a system of equation with no solutions, lines are parallel, same slope, different y-intercept

dependent system - a system of equation with infinite solutions

consistent dependent - a system of equation with at least one solution, infinite solutions, same slope, same y-intercept

consistent independent - one solution, lines intersect once, different slope, same y-intercept

4.7 Systems of Inequalities

Is the point (x, y) a solution to the inequality (a > b)

5.1 Polynomials and Functions

Polynomial is any expression in the form ax^2 + bx - c
where 'a' is a real coefficient of x and n is any nonnegative integer.

The degree of a polynomial is the highest degree of the term for the one above the greatest (8)

5.2 Addition and Subtraction of Polynomials

(yxy^2 - 6x^2 y^2 + 5x^3 y^2) - (2x^2 y^2 - 5xy^2 - 18x^3 y)

The additive inverse of a polynomial function 'a' is '-a' because a + (-a)=0

5.3 Multiplying Polynomials

multiplying binomials:
(2a+1)(4a^2 -2a+1)

special products:
(a-b)(a+b) = a^2 + ab -ab -b^2
= a^2 - b^2

squaring a binomial:
(a+b)^2 = (a+b)(a+b)

cubing a binomial:
(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3
(a-b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3

5.4 Factoring

(1.) Factor out the greatest common factor (gcf)

(2.) How many terms does the expression have?

   - if 2 terms: is it a difference of 2 squares?
a^2 - b^2 = (a - b)(a + b)

- if 3 terms: is it a trinomial square?

a^2 + 2ab + b^2 = (a + b)^2
a^2 -2ab + b^2 = (a - b)^2

- if 4 terms: a^2 + 2ab + b^2 - c^2 = (a + b)^2 - c
(1.) (a + b - c)(a + b + c)
(2.) 9y^4 - 15y^3 + 3y^2 (this is the same as dividing all by the gcf)
3y^2(3y^2 - 5y + 1)
6.7 Problem Solving: Rational Equations

(I.) rate x time = work done

6.6 Solving Rational Equations
In order to get rid of a denominator, a common denominator is needed.
(1.) Find excluded values (set all denominators equal to zero and solve)
(2.) Find LCD for all factors
(3.) Multiply all numerators by LCD
(4.) Simplify all Factors
(5.) Solve for variable
(6.) Make sure that any solution are not an excluded value.

5.5 More Factoring
(see above (5.4))

5.6 General Factoring Strategies
(1.) Factor out GCF
(2.) How many terms?

2 terms:
Difference of two squares:
a^2 - b^2 = (a-b)(a+b)
Difference of two cubes:
a^3 - b^3 = (a-b)(a^2 -ab + b^2)
Sum of two cubes:
a^3 +b^3 = (a+b)(a^2 -ab + b^2)

3 terms:
Trinomial square:
a^2 + 2ab + b^2 = (a+b)^2
a^2 - 2ab + b^2 = (a-b)^2

4 terms
(xw + xz)+(yw + yz)

5.7 Solving by Factoring

(x – 3)(x – 4) = 0

Set both equal to zero:
x – 3 = 0   or   x – 4 = 0

x = 3  or  x = 4
x = 3, 4

5.8 Problem Solving Using Factoring

'is' - 'equal sign'
'greater than' - 'addition'
'less than' - 'subtraction'

(use four step method)

6.1 Multiplying and Simplifying Rational Expressions

6.2 Addition and Subtraction of Rational Expressions

6.3 Rational Expressions

6.4 Dividing Polynomials

6.5 Synthetic Division

Synthetic division can only be used your divisor is in the form 'k -a' where 'a' is any number and 'k' is the variable.

6.8 Formulas

1/R = 1/r sub1 + 1/R sub2
R(r sub1)(r sub2)/R = R(r sub1)(r sub2)/r sub1 + R(r sub1)(r sub2)/r sub2
(r sub1)(r sub2) = R(r sub2) + R(r sub1)
subtract R(r sub2) from each side
(r sub1)(r sub2) - R(r sub2) = R(r sub1)
(r sub2)(r sub1 - R) =R(r sub1)6.9 Direct and Indirect Variation

Direct variation: if "y" varies directly as "x", "y" is proportional to "x"

y = k/x
y/x = k

"k" is the constant of variation or constant of proportion.

Inverse Variation: if "y" varies inversely as "x" or "y" is inversely proportional to "x"

y=k/x or xy = k

7.1 Radical Expressions

"n" square root of "a" = radical sign
"a" is called the radical and "n" is called the index.

root 4 = the "principal" square root of 4(positive square root of 4)
("equals what positive number squared is 4")

"Find the square root of 121/49"
= +/- root(121/49)
=+/- 11/7

7.2 Multiplying and Simplifying

For any nonnegative numbers "a" and "b" and any natural numbers index "k"

the "k" root" of "a" times the "k" root of "b" = the "k" root of "ab"
8.5 Equations reducible to Quadratic type

x^6 + 6^3 + 9 = 0

let u = x^3
u^2 = (x^3)^2
u^2 = x^6

u^2 + 6u + 9 = 0
In any equation the principal square root cannot be equal to zero.

7.3 Operations with Radicals
rationalize the denominator (simplify)

7.4 More Operations with Radicals
(see above)


7.6 Solving Radical Equations
Solving Radical Equations: Introduction (page 1 of 6)
A "radical" equation is an equation in which at least one variable expression is stuck inside a radical, usually a square root.

The "radical" in "radical equations" can be any root, whether a square root, a cube root, or some other root. Most of the examples in what follows use square roots as the radical, but (warning!) you should not be surprised to see an occasional cube root or fourth root in your homework or on a test.

In general, you "solve" equations by "isolating" the variable; you isolate the variable by "undoing" whatever had been done to it.

For instance, given x + 2 = 5, you would solve by undoing the addition of the 2. That is, the addition undone by applying the opposite: subtraction:

In the same manner, given something like –3x = 12,you would solve by undoing the multiplication by applying the opposite operation; namely, division:

When you have a variable inside a square root, you undo the root by doing the opposite: squaring. For instance, given , you would square both sides:

7.7 Imaginary and Complex Numbers

Imaginary Numbers are numbers which contain the factor sqroot(-1) = i

Complex numbers are numbers in the form 'a + bi' where 'a' and 'b' are real numbers.

7.8 Complex Numbers and Graphing

|a + bi| = sqroot(a^2 + b^2)

7.9 More about Complex Numbers

If a + bi = c + di then a = c and b = d.

The complex conjugate of a + bi is a - bi

7.10 Solutions of Equations

8.1 Introduction to Quadratic Equations

Complete the square steps:
(1.) divide both sides by 'a' if 'a' does not equal 1
(2.) move 'c' to the other side
(3.) complete the square by finding 'c'
(4.) add 'c' to both sides
(5.) factor and simplify
(6.) take the square root of both sides
(7.) solve the remaining absolute value equation

8.2 Problem Solving Using Quadratics

(use four step method)

8.6 Formulas and Problem Solving
(The four step method):

* Can you state the problem in your own words?
* What are you trying to find or do?
* What are the unknowns?
* What information do you obtain from the problem?
* What information, if any, is missing or not needed?
The following list of strategies, although not exhaustive, is very useful.
* Look for a pattern.
* Examine related problems, and determine if the same technique can be applied.
* Examine a simpler or special case of the problem to gain insight into the solution of the original problem.
* Make a table.
* Make a diagram.
* Write an equation.
* Use guess and check.
* Work backward.
* Identify a subgoal.
* Implement the strategy or strategies in step 2, and perform any necessary actions or computations.
* Check each step of the plan as you proceed. This may be intuitive checking or a formal proof of each step.
* Keep an accurate record of your work.
* Check the results in the original problem. (In some cases this will require a proof.)
* Interpret the solution in terms of the original problem. Does your answer make sense? Is it reasonable?
* Determine whether there is another method of finding the solution.
* If possible, determine other related or more general problems for which the techniques will work.

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