A(n) ______ number has the form a+bi, where a does not equal 0 and b does not equal 0

imaginary

A _________ number has the form a+bi, where a does not equal 0 and b=0

real

A _____ _______ number has the form a+bi, where a=0 and b does not equal 0

pure imaginary

The imaginary unit i is defined as i=_________, where i^2=___________

square root -1, -1

When a is a positive real number, the _____ _____ root of -a ois defined as square root -a=square root ai

principal square

The numbers a + bi and a-bi are called _________ ________ and their product is a real number a^2 + b^2

complex conjugates

what is the complex conjugate of root 6

root 6

what is the conjugate of i

-i

how would you do

2/1+i)-(3/1-i

do (2[1-i])-(3[1+i])

_______________

(1+i)(1-i)

when writing an imaginary quotient in standard form, what cannot be in the denominator

i, rationalize it

how would you do

i^44+i^150-i^74-i^109+i^61

divide each power by 4, is remainder is 1, result is i, if the remainder is 2, result is -1, if remainder is 3, result is -i, if remainder is 0, result is 1

x^2+1 has ____ real solutions

no

what is the pattern for imaginary numbers starting with i^1 and ending with I^4, before repeating itself over and over….

i (or, root -1), -1, -i, 1

what is standard form for complex numbers

a+bi

every complex number has a ______ number (a) and an _____ number (b)

real, imaginary

a+bi

if a=0, what number do you have?

pure imaginary

what is the complex conjugate of (3+2i)

(3-2i)

- same exact term with a sign change in the middle*
- things will cancel*

how do you rationalize 3+2i/6-i

multiply top and bottom by 6+i

when writing complex numbers in standard form, what do you ALWAYS do first

convert to imaginary

ex) root -3 times root -12

=root 3i times root 12i

=-6

what 3 solutions do you test first with synthetic division

-1,1,2

what does upper bound mean

when you plug in a number into synthetic division and you get all positive numbers as a result, that will be your highest root

imaginary roots come in _______, meaning a cubic function will have 2 roots with possible combinations ____…….

pairs,

3 real, 0 imaginary

1 real, 2 imaginary

NEVER 2 real, 1 imaginary

True or False, i is a variable

False- it is not a variable, but treat it like it is when performing operations

a set of ordered pairs

relation

The graphs of all polynomial functions are _______, which means that the graphs have no breaks, holes, or gaps

continuous

The ________ __________ _____ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function

leading coefficient test

A polynomial function of degree n has at most ___real zeroes and at most _____ turning points

n, n-1

When x=a is a zero of a polynomial function f, the following three statements are true:

a) x=a is a ______ of the polynomial equation f(x)=0

b) _____ is a factor of the polynomial f(x)

c) (a,0) is an _____ of the graph of f

zero, (x-a), x-intercept

When a real zero of a polynomial function is of even multiplicity, the graph of f _________ the x-axis at x=a, and when it is of odd multiplicity, the graph of f ______ the x-axis at x=a

bounces (touches) crosses

A factor (x-a)^k, k>1, yields a _______ _________ x=a of ______ k

repeated zero, multiplicity

A polynomial function is written in _______ form when its terms are written in descending order of exponents from left to right

standard

The _______ _________ Theorem states that if f is a polynomial function such that f(a) does not = f(b), then, in the interval [a,b], f takes on every value between f(a) and f(b)

Intermediate Value

how does a quartic graph behave

both sides go up/down

all polynomial functions are ____ and _____

continuous, curvy

are absolute value functions polynomials?

no- they are not curvy

describe the behavior of the following graph

f(x)=7x^2-x^7

left up, right down (be careful with highest degree)

zeros of a polynomial are also called …..

solutions, factors, x-intercepts, roots

what are points of inflection

where concavity changes

how many points of inflection can an equation have

degree-2

intermediate value theorem is also called ____________ ______________

existence theorem

what is a rational number?

irrational?

- fraction using integers (4/7, 10) –decimal will repeat or terminate
- decimal that will never repeat nor terminate (pi, root2)

how do you find roots of f(x)=x^3-x^2+2

factors of p/ factors of q

+1, -1, +2, -2/+1, -1

test -1,1,2

if f(a) (y value) <0 and f(b) >0 then in between a and b exists an _____

x-intercept

Linear, constant, and squaring functions are examples of ____ functions

polynomial

you can only use synthetic division with __ functions

linear

A polynomial function of x with degree n, must have a __________ ___________ degree n and a _______ x

positive integer, real

A ______ function is a second degree polynomial function, and its graph is called _________

quadratic, parabola

The graph of a quadratic function is symmetric about its ____________________.

axis of symmetry

When the graph of a quadratic function opens downward, its leading coefficient is _________ and the vertex of the graph is a _________

negative, maximum

when the graph of a quadratic function opens upward, its leading coefficient is ______ and the vertex is a _______

positive, minimum

describe the graph of g(x)=(3x)^2 +1

horizontal shrink, up 1

how do you find the vertex of an equation

-b/2a

plug result in for y part

quadratic functions-

degree 2 polynomial

graph: parabola

what is the standard quadratic function

f(x)=ax^2+bx+c

a cannot =0, c y intercept

what is the standard form for the equation of a parabola

f(x)=a(x-h)^2+k

a cannot =0

vertex (h,k)

how do you complete the square

make sure function is in standard quadratic form

group variable terms

make a=1 or factor out a

add (1/2b)^2

balance function (subtract ^ on outside)

factor trinomial (perfect square trinomial=2 same binomials)

rewrite in standard for for equation of parabola*REMEMBER–WHEN BALANCING, IF YOU FACTORED OUT AN A BEFOREHAND, CONSIDER THE ACTUAL VALUE TO SUBTRACT ON THE OUTSIDE*

equation of a parabola= f(x)=2(x+3/4)^2 -65/8 what is vertex? axis of symmetry? x intercepts? y intercepts? image to y-intercept?

(-3/4, -65/8)

x=-3/4 (remember to include x=)

x intercepts- use quadratic formula or plug in 0 for y (original equation)

y intercepts- plug in 0 for x (original equation)

imagine- (-3/2, -7) (same y value, cross over axis)

what is the position function? (feet)

s(t)=-16t^2+Vot+So

- 16 is constant
- V is initial velocity
- S is initial position in feet
- t is time

what is the position function (meters)

s(t)=-4.9t^2+Vot+So

In the Division Algorithm, the rational expression r(x)/d(x)is _____ because the degree of r(x) is less than the degree of d(x)

proper

In the Division Algorithm, the rational expression f(x)/d(x) is _____ because the degree of f(x) is greater than or equal to the degree of d(x)

improper

cubic/linear is an example of a _________ rational expression

improper

an alternative method to long division of polynomials is calles _______ ______, in which the divisor must be of the form x-k

synthetic division (only when divisor is linear)

The ______ Theorem states that a polynomial f(x) has a factor (x-k) iff f(k)=0

Factor

The _____ Theorem states that if a polynomial f(x)is divided by x-k, then the remainder is r=f(k)

Remainder

the _______ is divided by the _______ which gives you a _____ with a ________

dividend, divisor, quotient, remainder

dividend=quotient*divisor+remainder

division algorithm

dividend/divisor=quotient+remainder/divisor

alternative (division) algorithm

when you test a positive number and get all positives in quotient, you have a __________ ____________

Upper bound

What are extrema and extremum

Relative max and mins

Critical points include

Relative max and mins and intercepts

Intermediate value theorem is an ______ theorem

Existence