Algebra II

Question 1

Always check your solutions in the original equations because…

Answer 1

Manipulating the original equations sometimes causes extraneous (false) solutions

This includes squaring and absolute value stuff

Question 2

What is the end goal of completing the square

Answer 2

Find solutions to an equation

By first creating a perfect square

ex. of a perfect square: (x-3)² = x²-6x+9

Question 3

Anytime you are dealing with algebraic inequalities

Answer 3

Check everything on a numberline!

Linear or quadratic…

And check the “equals to” for the domain as well

Question 4

When factoring cumbersome quadratic equations. Here is a good trick

Answer 4

Just use the quadratic formula, but don’t simplify the solutions of the formula to an integer, leave it as the most simplified fraction, from there, you can “back-track” that solution to create the equivalent ax+b…

Question 5

Completing the square process

start: 3x² + 10x - 8 = 0
end: x = 2/3 or -4

Great practice is deriving the quadratic equations from

ax² +bx +c = 0

Answer 5

1. Divide as appropriate to ensure the coefficient of the highest power is = to 1
2. Move the constant term to the other side
3. Take half of the value of the coefficient on the first-degree term of the variable, square the result of the halving, then add to both sides.

Say you are halfing and squaring b… The half part is what creates the perfect sqare (x+b/2)²

4. Factor the side of the equation that’s a perfect square trinomial (write it simplified like (x-3)²)
5. find the square root of both sides (be sure to put the plus/minus in front of the non-variable square root)
6. Solve for the variable

Question 6

Completing the square on two variable polynomials

start: x² + 6x + 2y² - 8y + 13 = 0
end: (x+3)² + 2(y-2)² = 4

Answer 6

1. Combine all of the constants to the other side
2. Divide as necessary to make the highest “x” coefficient = 1
3. Factor out the constant of the other variable and leave it
4. Complete the squares on both sections and you have the result
5. The result is often an equation for a conic section

Question 7

How many solutions are in a cubic equation

Answer 7

Up to 3

Question 8

What are quadratic-like trinomials

Answer 8

ax²ⁿ + bxⁿ + c = 0

Solve by factoring them, then find the cubed root

Question 9

When are you going to start jumping to factoring by grouping?

Answer 9

When you see a cubic polynomial or higher power, that can’t be simplified by factoring out the higher power from all terms

Question 10

How to solve quadratic inequalities

Answer 10

Solve for solutions as normal, remember to flip the inequality as necessary.

Draw a number line with the solutions on it

Test the numbers to find out if the solution is neg or pos in that particular domain

If the result matches the inequality, then that is the domain of the solution

Question 11

How to solve rational inequalities

ex. (x-2)/(x+6) < 0

Answer 11

Solve for zeros, make a sign line. Then find the domain that satisfies the inequality

Question 12

When solving rational equations, what is the most likely method to accomplish it

Answer 12

Multiply out by the LCD (least common denominator)

! watch out for erroneous solutions (check answers)

Question 13

Remember how you can use proportions so find solutions

a/b = c/d

ad = bc

and : b/a = d/c

Answer 13

Multiply it out and solve for solutions

Question 14

What to do if you have a radical term in the equation and need to solve for roots

Answer 14

Square it (or cube or whatever), and check solutions!!!

Question 15

What do you do if the equation has 2 radicals that can’t be combined

start: sqrt(3x +19) - sqrt(5x -1) = 2
end: x = 2, 34

(plugging in for a test shows that 34 is extraneous)

Answer 15

1. Move the radicals so that only one appears on each side
2. Square both sides of the equation
3. Isolate the remaining radical on one side
4. Simplify the coefficents by dividing
5. Square both sides again
6. Solve for the solutions

Question 16

What to do if you have a negative exponent

Answer 16

Write it out as a reciprocal and solve

Or, factor out the GCF ex:

3x^-3-5x^-2 = 0

is

x^-3(3-5x) = 0

When you solve for solutions with negative exponents, remember that you can never have zero in the denominator, so they won’t be solutions

Question 17

Something to do if you have fractional exponents

Answer 17

Find a GCF that is the variable to whatever fractional power, and factor it out

Question 18

Quick way to tell the highest power of a polynomial based on the graph

Answer 18

The number of humps + 1 is the highest order

line (x) = 0 humps

quadratic (x²) = 1 hump

cubic (x³) = 2 humps

quartic (x⁴) = 3 humps

Question 19

A radical graph looks like:

Answer 19

Question 20

A rational graph looks like:

Answer 20

y = 5 / (x-3)

Question 21

Graph of a exponential curve:

y = ab^x

Answer 21

y = 5(3^x)

They are always upward

Question 22

Graph of a logarithmic curve

Gerneral form: y = log_bx

Answer 22

Always downward

Question 23

Graph of absolute value

Answer 23

Question 24

Concerning roots from fractional exponents

Answer 24

They can be like even exponents, where the + and - root exists

Question 25

Even vs. odd functions

Answer 25

Even: you get the same y value for +x and -x Odd: you get opposite y values for +x and -x Even is symmetrical about the y axis, odd is symetrical about the x axis

Question 26

What does it mean for a function to be one-to-one

Answer 26

If you calculate exactly one output value for every input value and exactly one input value for every output value. This means that the function has an inverse Functions with even exponents are automatically not one to one

Question 27

Vertical line test vs. Horizontal line test

Answer 27

All "functions" pass the vertical line test Only one-to-one functions pass the horizontal line test

Question 28

Piecewise function

Answer 28

One function that contains multiply functions for different domain ranges

Question 29

Power vs. exponential

Answer 29

Power: y = x² Exponential: y = 2^x (much faster increase)

Question 30

What is composition

Answer 30

f(g(x))... stuff like that

Question 31

What is the difference quotient

Answer 31

Foundation of the derivative

Question 32

Inverse functions (f^-1(x))

Answer 32

Definition: f (f^-1(x)) = x and f^-1(f(x)) = x

Question 33

How to solve for an inverse function

Answer 33

1. Replace f(x) with y 2. Swap all x's and y's 3. Simplify until you get one y! (try grouping) 4. Once you have "y=", that is the inverse function

Question 34

Know the difference between relative and absolute maximums / minimums

Question 35

Rational Root Theorem

Answer 35

factors of the constant divided by factors of the coeffiecient of the highest power What ever result works with synthetic division is a root, but remember when expanding through factoring that it is x +/- the opposite sign of the root

Question 36

Descartes' Rule of Signs (Part 1)

Answer 36

A polynomial with xⁿ as the highest degree will have at most, n roots. Count the number of times the sign changes in the polynomial, call that p, and that will equal the maximum number of positive roots of the polynomial. This includes the constant's sign!, and need to be in order If the number of positive roots isn't p, it is p-2, p-4 ... mult of 2.

Question 37

Descartes' Rule of Signs (Part II)

Answer 37

A polynomial with xⁿ as the highest degree will have at most, n roots. Evaluate the polynomial as f(-x). Count the number of times the sign changes in the polynomial, call that q, and that will equal the maximum number of negative roots of the polynomial. This includes the constant's sign! and need to be in order If the number of positive roots isn't q, it is q-2, q-4 ... mult of 2.

Question 38

When dividing polynomials, what happens when synthetic division leaves a remainder

Answer 38

Rewrite the polynomial. A few changes... At the end, write the remainder as a numerator with the divisor as the denominator. Then put the constant to the left, then add the power x's. The final result will have a weird remainder, a constant, and a polynomial that is one degree lower than the original.

Question 39

Property of the Remainder Theorem

Answer 39

If a polynomial f(x) is divided by the binomial x-c, the remainder of the division is equal to f(c) Say (x+2) is the divisor. If the remainder is -4 then f(c) = r or f(-2) = -4 This is valuable if doing the whole calculation raw is too cumbersome (great for computers)

Question 40

Set notation with u and upside down u

Answer 40

Question 41

How to quickly determine the horizontal asymptote of a polynomial divided by another polynomial

Answer 41

If the highest degrees of each polynomial are equal, find the ratio between the two coefficients. This will create a y value that is the intercept for the asymptote line

Question 42

Quickly determine the vertical asymptotes of two polynomials divided by eachother

Answer 42

When the denominator equals zero

Question 43

Quickly determine the oblique (slant) asymptotes of a polynomial divided by another polynomial

Answer 43

The degree of the polynomial in the numerator must be exactly one degree higher than the denominator. Divide it out... The binomial result is the equation for y =

Question 44

Polynomial long division

Answer 44

Question 45

Removable discontinuity vs. nonremovable discontinuity

Answer 45

Question 46

How to determine a removable discontinuity in a rational function

Answer 46

Factor the numerator and the denominator. What ever term cancels out, the x value that makes that term zero is a removable discontinuity. Example:

Question 47

What does it mean if a function has a removable discontinuity

Answer 47

A limit exists Solve for the removable discontinuity. Simplify the rational function, and plug in the root of the removable discontinuity and solve. The result is the limit

Question 48

How to tell if approaching infinity or zero

Answer 48

zero = 1/x Infinite is if the numerator polynomial is of a higher degree than the denominator A constant is if the numerator and denominator polynomials are of the same degree

Question 49

Approx value of "e"

Answer 49

2.718

Question 50

General notation for log y = log_b n is... y = ln n is...

Answer 50

b^y = n e^y = n

Question 51

Laws of logs

Answer 51

Question 52

ln(e) = ...

Answer 52

1

Question 53

Change of base

Answer 53

Question 54

What are the four conic sections

Answer 54

Circle, parabola, ellipse, hyperbola

Question 55

Wordy definition for a parabola

Answer 55

All the points that fall the same distance from the focus and the directrix

Question 56

General Equation for a Parabola

Answer 56

Left / Right: (y-k)² = 4a(x-h) Up / Down: (x-h)² = 4a(y-k) Vertex is (h,k), 4a positive = right / up, 4a negative = left / down, the larger 4a is, the wider it is, focus is (0, a) or (a, 0), directrix is x/y = -a

Question 57

Parabola depiction

Answer 57

Question 58

Standard Equation for a circle

Answer 58

(x-h)² + (y-k)² = r²

Question 59

Wordy definition of an ellipse

Answer 59

All the points where the sum of the distances from the points to the two foci are constant

Question 60

Standard equation for an ellipse

Answer 60

Center (h,k) Major axis, minor axis a is half of the L/R axis b is half of the U/D axis

Question 61

Two types of ellipse

Answer 61

Question 62

Wordy definition of a hyperbola

Answer 62

All the points that the difference of the distances between the two foci are constant

Question 63

Standard Equation for a Hyperbola

Answer 63

Question 64

Hyperbola depiction

Answer 64

Question 65

Quick ways to tell what conic section it is

Answer 65

Parabola - only one squared variable Circle - two squared variables, no fractions Ellipse - sum of two fractional squares Hyperbola - difference of two fractional squares

Question 66

Quick way to tell what conic section when not in standard form

Answer 66

for: Ax² + By² + Cx + Dy + E = 0 If A = B: circle If A != B, and same sign: ellipse If A and B are different signs: hyperbola If A or B is zero: parabola

Question 67

Image with all conic section standard equations

Answer 67

Question 68

Three possible scenarios with a system of linear equations

Answer 68

One solution (where the lines intersect) Infinite number of solutions (same line) No solution (lines parallel)

Question 69

Two most common ways to solve a system of linear equations

Answer 69

Elimination and substitution Eliminiation you just multiply, then add (each equation can be multiplied by a different constant)

Question 70

Overall guide to substitution

Answer 70

Solve one of the equations for one of the variables x or y Substitute the value of the variable into the OTHER equation

Question 71

What form is required for Cramer's Rule for solving a system of linear equations

Answer 71

a₁x + b₁y = c₁ a₂x + b₂y = c₂ and... d = a₁b₂ - b₁a₂

Question 72

How to solve a system of 3 linear equations (this can be done for any system of equations as long as there are as many equations as there are variables)

Answer 72

1. Pick any two pairs of equations from the system. (Actually make two sets of systems of equations) 2. Eliminate the same variable from each pair using the Addition/Subtraction method. 3. Solve the system of the two new equations using the Addition/Subtraction method. (End up solving for one variable, then plug into one of these equations to solve for the other variable. Remember, not the original equation) 4. Substitute the solution back into one of the original equations (3 variable) and solve for the third variable. 5. Check by plugging the solution into one of the other three equations.

Question 73

Overall Point of Decomposition / Expansion of rational polynomials

Answer 73

To get to a simpler term so you can integrate or inverse laplace

Question 74

Overall Process For Partial Fraction Expansion

Answer 74

If its not 1 degree lower, do long polynomial division to get it to that point (be sure to keep track of the results/ remainders) Then immediately try to factor the denominator. This may result in a quadratic term in the denominator. Use synthetic division to get roots. Notice that the first and third cases are really special cases of the second and fourth cases respectively. Check out khan academy and http://archives.math.utk.edu/visual.calculus/4/partial\_fractions.1/

Question 75

When solving nonlinear systems of equations... Some dead giveaways

Answer 75

If you have a complex root, they don't intersect (no solution) If you have a double root (the same solution appears twice), they are tangent to eachother

Question 76

Powers of "i"

Answer 76

i = i i² = -1 i³ = -i i⁴ = 1

Question 77

Standard format of complex number

Answer 77

a +bi

Question 78

adding complex numbers and subtracting complex numbers

Answer 78

(a+bi) + (c+di) = (a+c) + (b+d)i (a+bi) - (c+di) = (a-c) + (b-d)i

Question 79

multiplying complex numbers

Answer 79

(a + bi)(c + di) = (ac - bd) + (ad + bc)i You can also just FOIL this and add like terms and change i² to -1. Gives the format

Question 80

What is a complex number's conjugate?

Answer 80

it is the same complex number, just an opposite sign of the imaginary part. The product of these two equals a real number: (a + bi)(a - bi) = a² + b²

Question 81

How to divide complex numbers

Answer 81

Question 82

Matrix dimension notation

Answer 82

rows X columns

Question 83

Square matrices, zero matrices, identitiy matrices

Answer 83

Even rows and columns, all zeros, and all zeros except ones going at a diagonal from the top left to bottom right

Question 84

How to add or subtract matrices

Answer 84

Have to be the exact same size, add or subtract the corresponding position

Question 85

Scalar multiplication fo a matrix

Answer 85

Just multiply each element by the constant

Question 86

The required dimensions to multiply two matrices

Answer 86

Say two matrices of dimensions [a, b] \* [c, d] b must equal c, and the resulting dimensions are [a, d]

Question 87

Mechanics of matrix multiplication

Answer 87

Question 88

What is an additive inverse matrix

Answer 88

When you add it to the original matrix, you get a matrix of all zeros (all matrices have them, but this is not the case for multiplicative inverses)

Question 89

What happens when a matrix is multiplied by an identity matrix (all zeros with a one diagonal)

Answer 89

Maintains the original matrix

Question 90

What does multiplicative inverse mean

Answer 90

The matrix, A, is the multiplicative inverse of the other (denoted A^-1), if the result of the two is all zeros with a diagonal of ones

Question 91

Quick way to get an inverse of a 2 x 2 matrix

Answer 91

(Multiply out the scalar)

Question 92

There are ways to get the inverse of larger matrices, but it's fairly complicated. It involves determinates and cofactors

Question 93

How to solve a system of equations with matrices (great for computers and calculators)

Answer 93

Rules 1. All the equations must be linear. With as many equations as there are variables. 2. Create a square coefficient matrix, A. Create a constant matrix, B. B contains all of the constants, sign wise, as if they were on the other side of the equation than the variables. 3. Find the multiplicative inverse of matrix A. 4. Multiply A^-1 x B. 5. Result is a column of the solutions.

Question 94

How to divide matrices

Answer 94

Mutiply one matrix by the inverse of another

Question 95

What is the difference between a sequence and a series

Answer 95

A sequence is a list of numbers, a series is the sum of the numbers in that list. The true definition of a sequence is a function whose domain cositst of positive integers

Question 96

Sequence notation

Answer 96

Example: {a_n} = {n² -1} a₁₀ = 99

Question 97

How to create an alternating sequence (pos to neg to pos forever)

Answer 97

{-1ⁿ x n}

Question 98

Arithmetic sequences

Answer 98

Constant increase, d, between the integers formula a_n = a_n-1 + d

Question 99

Geometric sequence

Answer 99

Each term is different from the one that follows it by a common ratio gn = rg_n-1

Question 100

Summation notation

Answer 100

Add all numbers genernated from the function a_k, starting at the k value all the way to the n value

Question 101

Sum of the first n terms of an arithmetic sequence

Answer 101

Sum = n/2 (a₁+a_n)

Question 102

Sum of the first n terms of a geometric sequence

Answer 102

r is the ratio, n is the number of terms

Question 103

Sum of an infinite geometric series

Answer 103

r is the ratio, g₁ is the first term

Question 104

How many subsets are in a set

Answer 104

if a set A has n elements: 2ⁿ subsets

Question 105

What is a union of sets

Answer 105

Combining

Question 106

What is an intersection of sets

Answer 106

The resulting set is only what both sets had. Notated as an upside down U

Question 107

Relational equation of union and intersection

Answer 107

Question 108

Interesting factorial property

Answer 108

n! = n\*(n-1)!