WGU C278 - College Algebra (in progress as I study)
20²
400
Linear function
A function represented by an equation of the form y=mx+b. The domain is the set of all real numbers DE=(-∞,∞)
Product Rules for Exponents
Add the exponents together
Conditional Equation
An equation with a finite number of solutions
Identity (Equation)
An equation with an infinite number of solutions
Contradiction (Equation)
An equation with no solutions
Standard Form (equation)
Ax+By=C
Factor trinomial with leading coefficient 1
Find two factors of the constant term whose sum in the coefficient of the middle term. (if they do not exist, the trinomial is not factorable) Example: x²+11x+30 = (x+5)(x+6)
Finding the Roots
Finding the Solutions to an equation
Rules for Dividing Positive and Negative Real Numbers
For positive real numbers a and b, The quotient of two positives is positive: ab=+ab. The quotient of two negatives is positive: −a−b=+ab. The quotient of a positive and a negative is negative: −ab=a−b=−ab. In summary: The quotient of numbers with like signs is positive. The quotient of numbers with unlike signs is negative.
18²
324
19²
361
∈
"an element of"
∣
"such that"
{x∣x∈ℤ}
"the set of all x such that x is an element of the set of integers"
{x∣x is an even integer}
"the set of all x such that x is an even integer"
Vertical line
(#,0) x=a slope is 0
Negative base number in parenthesis
(-x)² = (-x)(-x) With parenthesis the base is negative
Horizontal line
(0,#) y=b slope is 0
Slope of the line (equation)
(Slope (m) = Change in Y ÷ Change in X = Rise ÷ Run and x₁ ≠ y₁)
Quotient Rule for Exponents
(top exponent) minus (bottom exponent)
FOIL Method
*F* irst terms *O* utside terms *I* nside terms *L* ast terms
Solve Inequalities
--Multiply or Divide by a negative number results in an OPPOSITE inequality sign. --All else keeps the inequality same the same.
Function (ƒ)
-A relation in which each domain element has exactly one corresponding range element. -A relation in which each first coordinate appears only once. -A relation in which no two ordered pairs have the same first coordinate.
Subtracting polynomials
-Change the signs of the term being subtracted (positive to negative and vice versa) -Combine like terms (written with highest degree aka. exponent first)
Monomial
-n is a whole number -k is a real number -No variables or positive exponents in the denominator. -No negative exponents in the numerator. -No fractional exponents (only whole #s)
Negative base number without parenthesis
-x² = (-1)x² Without parenthesis the base is not negative but is preceded by a -1
Factoring out the GCF (find a monomial that is the GCF of a polynomial)
1. Find the variable(s) of highest degree and the largest integer coefficient that is a factor of each term of the polynomial. (This is one factor) 2. Divide this monomial factor into each term of the polynomial resulting in another polynomial factor. (if the first term is negative, always do a -GCF)
Perfect Square Trinomials (Square of a binomial sum)
1. the first term is squared (first of the trinomial) 2. the coefficients of the two are multiplied and then doubled (second of the trinomial) 3. the second term is squared (third of the trinomial)
11²
121
12²
144
13²
169
14²
196
15²
225
Rational Number Examples
23,71,−53,2710, and 3−10 Each is in the form ab where a and b are integers and b≠0. 13/4 and 2.33 are also rational numbers. They are not in the form ab, but they can be written in that form: 13/4=7/4 and 2.33=233100=233100
16²
256
17²
289
Polynomial with *two* terms (after simplification)
Binomial Example: 4x-10 (1st-degree binomial)
Adding polynomials
Combine like terms (written with highest degree aka. exponent first)
Formula for Distance
D=RT (Distance = Rate × Time)
Linear polynomial
Degree 0 or 1 Example: 3x+8 (1st-degree binomial, leading coefficient 3)
Quadratic polynomial
Degree 2 Example: 5x²-3x+7 (2nd-degree trinomial, leading coefficient 5)
Cubic polynomial
Degree 3 Example: -10y³ (3rd-degree monomial, leading coefficient -10)
Whole number relationships
Every one of these numbers is also a: -integer -rational number -real number
Integer relationships
Every one of these numbers is also a: -rational number -real number
Irrational number relationships
Every one of these numbers is also a: -real number
Rational number relationships
Every one of these numbers is also a: -real number
Natural number relationships
Every one of these numbers is also a: -whole number -integer -rational number -real number
Real number line
Every rational and irrational number has a corresponding point on this line.
Absolute Value Examples
Example 1: Absolute Value |5|=5 |−6|=−(−6)=6 |π|=π |0|=0 ∣∣∣−34∣∣∣=−(−34)=34 −|−5.2|=−5.2 If |x|=6, what are the possible values for x? x=6 or x=−6 because both |6|=6 and |−6|=6. We say that {−6,6} is the solution set of the equation. If |x|=−4, what are the possible values for x? There are no values of x for which |x|=−4. The absolute value can never be negative. The solution set is the empty set, ∅.
Rational Numbers
This is any number that can be written in the form a/b where a and b are integers and b≠0. (The letter ℚ represents the whole set)
Real numbers
Formed with the set of rational and irrational numbers. (The letter ℝ represents the whole set)
Formula for Simple Interest
I=PRT (Interest = Principle × Rate × Time) I=PR (Interest = Principle × Rate) if just one year
Factor Theorem
If x=c is a root of a polynomial equation in the form P(x)=0, then x-x is a factor of the polynomial P(x).
Double root (root of multiplicity two)
If you have a double factor, then the solution is called ____. Equation: (x+1)(x+1)=0 Answer = -1 (doubled)
Factoring Perfect Square Trinomials
In a perfect square trinomial, both the first and last terms of the trinomial must be perfect squares. If the first term is of the form x² and the last term is of the form a², then the middle term must be of the form 2ax or −2ax. x²+2ax+a² = (x+a)² x²-2ax+a² = (x-a)²
Consecutive Integers
Integers are _______ if each is 1 more than the previous integer. Three of these can be represented by n, n+1, n+2. Example: 5,6,7
Polynomial with *one* term (after simplification)
Monomial Example: 15x³ (3rd-degree monomial)
Power Rule
Multiply the exponents
Evaluation of Polynomials
P is the polynomial and x is the variable used in the polynomial. P(3) means 3 is substituted for all x's
Formula for Profit
P=S-C (Profit = Selling Price - Cost)
Order of Operations
PEMDAS (Please excuse my Dear Aunt Sally) 1. Parethesis (groups) 2. Exponentials 3. Multiplication and Division, left to right 4. Addition and Subtraction, left to right
Not factorable (irreducible or prime)
Polynomials that do not have a GCF (greatest common factor)
Power Rule for Products
Raise each factor to that power
Power Rule for Quotients (Fractions)
Raise the numerator and denominator to that power
0
Rational Number, Integer, Whole Number
Rule for Negative Exponents
Reciprocal of the base.
Slope
Rise ÷ Run
Relation
Set of ordered pairs of real numbers
Polynomial with *three* terms (after simplification)
Trinomial Example: -x⁴+2x-1 (4th-degree trinomial)
rules for order of operations
Simplify within symbols of inclusion (parentheses, brackets, braces, fraction bar, absolute value bars) beginning with the innermost symbols. Find any powers indicated by exponents or roots. Multiply or divide from left to right. Add or subtract from left to right.
Parallel lines
Slope is identical y=2x+1 y=2x+3
Perpendicular lines
Slope is the negative reciprocal y=½x+1 y=-2x-3
16⋅3÷23−(18+20)
Solution: 16⋅3÷=====23−(18+20)16⋅3÷23−3816⋅3÷8−3848÷8−386−38−32Add within the parentheses.Evaluate the exponents.Multiply or divide from left to right.Add or subtract from left to right.
Difference of Two squares
Sum of the same binomial terms *but* one is addition and one subtraction.
Symbols for Multiplication
Symbol Description Example ⋅ raised dot 4⋅7 ( ) numbers inside or next to parentheses 5(10) or (5)10 or (5)(10) × cross sign 6×12 or 12×6¯¯¯¯¯¯ number written next to variable variable written next to variable 8x xy
Domain (D)
The ___ of a relation is the set of all first coordinates in the relation.
Range (R)
The ___ of a relation is the set of all second coordinates in the relation.
Leading coefficient (Coefficient of the polynomial)
The coefficient of the term of the largest degree. (aka the coefficient of the term with the largest exponent)
Domain axis
The horizontal axis (x-axis) is called this
Degree of a polynomial
The largest of the degrees of its term after like terms have been combined. With multiple exponents - use the largest one AFTER all like term are combined.
Hypotenuse
The longest side of a right triangle.
Range axis
The vertical axis (y-axis) is called this
Whole Numbers
These are counting numbers starting with the number 0. {0,1,2,3,4,5,6,...} (The letter 𝕎 represents the whole set)
Natural Numbers
These are counting numbers starting with the number 1. {1,2,3,4,5,6,...} (The letter ℕ represents the whole set)
Integers
These are numbers that can be represented on a number line, no decimals. {..., −4,−3,−2,−1,0,1,2,3,4,...} (The letter ℤ represents the whole set)
Consecutive Even Integers
These integers are _______ if each is 2 more than the previous even integer. Three of these can be represented by n, n+2, n+4. Example: 24, 26, 28
Consecutive Odd Integers
These integers are _______ if each is 2 more than the previous odd integer. Three of these can be represented by n, n+2, n+4. Examples: 41,43,45
Variable
This is a symbol (generally a letter) that is used to represent an unknown number or any one of several numbers.
Irrational Numbers
This is any number that can be written as an infinite, non-repeating decimal. These cannot be written in fraction form.
System of equations Set of simultaneous equations
Two or more linear equations considered at one time.
Factor trinomial with leading coefficient other than 1
Use the FOIL method, followed by trial and error. -Find all various combinations of *F* and *L* -List all possible combinations in their respective *F* and *L* positions. -Only check the sums in the *O* and *I* positions, until you find the sum of the middle term. (if they do not exist, the trinomial is not factorable) Example: 6x²+23x7 = (3x+1)(2x+7)
Solve Equalities
What is done to one side of the equation must be done to the other
Applications (aka)
Word Problems (aka)
Irrational Number Examples
a. π=3.14159265358979... π has no repeating pattern in its decimal form. b. 2√=1.414213562... The square root of 2 has no repeating pattern in its decimal form. c. e=2.718281828459045... e is a number used in higher mathematics and engineering courses. d. 0.01001000100001... Even though there is a pattern to the digits, the pattern is not repeating.
term
an expression that involves only multiplication and/or division with constants and/or variables. Examples: 2x⁵, ²/₃x²y, 14, 5.6a
Second-degree trinomials (in the variable x)
ax²+bx+c (a, b, and c are real constants)
ac-Method of Factoring trinomials
ax²+bx+c 1.Multiply *a* & *c* 2.Find two integers whose product is *ac* and whose sum is *b*. (if none, then not factorable) 3.Rewrite the middle term *bx* using the two numbers from above as coefficients. 4.Factor by grouping the 1st two terms and the last two terms. 5. Factor out the common binomial factor. This will give two binomial factors.
Quadratic equations
ax²+bx+c=0 a, b, and c are constants and a≠0 (if a=1, then the answer is the 2 factors whose product is *c* and sum is *b*, check your positive/negative signs)
Exponent One (1)
a¹=a
Exponent Zero (0)
a⁰=1 0⁰=undefined
Pythagorean Theorem
c²=a²+b²
>
greater than
≥
greater than or equal to
Zero-factor Property
if ab = 0, then a=0 or b=0 or both. Before factoring out anything, get everything one one side of the equals side so it equals 0
Vertical line test
if any vertical line intersects the graph of a relation at more than one point, then the relation is NOT a function
Number Examples 2
integers. Solution: −2 and 0 are integers. rational numbers. Solution: −2,−1.1,−12,0,58, and 1.7 irrational numbers. Solution: −3√ and 1.7−−−√ are irrational numbers. −3√ is approximately −1.732 and 1.7−−−√ is approximately 1.304.
∩ (A∩B)
intersection symbol: meaning all elements belong to both. Think "and"
Coefficient of the monomial
k is called this
<
less than
≤
less than or equal to
Polynomial
monomial or the algebraic sum or difference of monomials
Degree of the monomial
n is called this With multiple exponents - add them together.
Perfect Square Trinomials (Square of a binomial difference)
the middle term is doubled
Consistent
two linear equations in two variables: have exactly one solution (the lines intersect at one point)
Inconsistent
two linear equations in two variables: have no solution (the lines are parallel)
Dependent
two linear equations in two variables: have an infinite number of solutions (the same line)
∪ (A∪B)
union symbol: meaning that the set belongs to either or to both. Think "or"
(x+a)²
x²+2ax+a²
Not factorable squares
x²+a ² = *Not* factorable (x-a)² = Factorable
(x-a)²
x²-2ax+a²
(x+a)(x-a)
x²-a²
Point-Slope Form (equation)
y-y₁=m(x-x₁)
Nonlinear function
y=2÷x-1 (x≠1) D=(-∞,1)∪(1,∞)
Slope-Intercept Form (equation)
y=mx+b (m is the slope and the y-intercept is (0,b))
function notation (formula)
ƒ(x)=mx+b
Number Examples
−3 : integer, rational number, real number 12 : rational number, real number 4 : natural number, whole number, integer, rational number, real number 10−−√ : irrational number, real number
