factor x^3+8

(x+2)(x^2-2x+4)

how do you factor a perfect cube

in the first (), take the cubed root of both variables in the equation

in the second (), first, put in whatever you need to get the leading coefficient, then put in the opposite of the product of the first (), then whatever you need to get the constant

is (x-1)(x^2+x+1) a product of linear factors

no, simplify the last trinomial by doing quadratic formula to get a product of linear factors

is x^2 + 49 factorable?

yes

x-7i) (x+7i

what is the rational zero test

p(constant)/q (leading coefficient

some factors of both of these things will present a zero of the equation

what are the possible root combinations for a cubed equation

all real, 2 imaginary and one real

what does lower bound say

you plug in a negative number and get alternating sums in return,

this will be your lowest zero

what is upper bound

you plug in a higher number and get all positive sums in return

this will be your highest zero

what is DesCartes’s Rule of signs

the number of positive real zeros=the number of sign variations or less than that by an even integer

the number of negative real zeros=the number of sign variations at

f(-x) or less than that by an even integer

what does irreducible over the reals mean

does not reduce into real factors

x^2-2x+10

write x^6-x^7 as a product of linear factors

(x)(x)(x)(x)(x)(x)(1-x)

The _______ _______ of _______ states that if f(x) is a polynomial of degree n (n>0), then f has at least one zero in the complex number system

fundamental, theorem, algebra

The _______ ________ _______ states that if f(x) is a polynomial of degree n (n>0), then f(x) has precisely n linear factors

linear factorization theorem

The test that gives a list of the possible rational zeros of a polynomial function is the ______ _______ Test

rational zero

If a+bi is a complex zero of a polynomial with real coefficients, then so is its ________, a-bi

conjugate

Every polynomial of degree n>0 with real coefficients can be written as the product of _______ and ______ factors with real coefficients, where the ___________ factors have no real zeros

polynomial, linear, quadratic

A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be _______ over the _______

irreducible, reals

The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called __________ __________ of ________

Descartes’s Rule, Signs

A real number b is a ______ bound for the real zeros of f when no real zeros are less than b, and is a ______ bound when no real zeros are greater than b

lower, upper

if 5i is a zero, what else is true

-5i is a zero

a set of ordered pairs

relation

what is a rational function

fraction with polynomials

how do you find the vertical asymptote

whatever makes the denominator 0

roots of denominator

x=

be sure to factor function first as asymptotes might cancel

how do you find the horizontal asymptote if numerator’s degree is < denominator’s degree

y=0

it is the x-axis

how do you find HA if the numerator’s degree=denominator’s degree

y=ratio of leading coefficients

f(x)=2x-5/4-x HA:y=-2

how do you find the HA if you numerator’s degree > denominator’s degree by 1?

by 2?

by 1: y=quotient slant asymptote

by 2: y=quotient parabolic asymptote

how do you find x intercepts and y intercepts of a rational function

x: roots of numerator (set=0) make sure to factor first in case things cancel

y: plug 0 in for x

NOTE- be sure to write as ordered pairs, (0,0) is not a y or x intercept

what is intermediate form? is it the same as undefined

0/0, no

slant asymptotes and parabolic asymptotes all begin with

y=

so do HA

VA begins with x=

functions of the form f(x)=N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial are called ______ _______

rational functions

when f(x)—>+/- infinity as x—>a from the left or right, x=a is a _______ ________ of the graph of f

vertical asymptote

when f(x)—>b as x—>+/- infinity, y=b is a ______ ______ of the graph of f

horizontal asymptote

for the rational function f(x)=N(x)/D(x), if the degree of N(x) is exactly one more than the degree of D(x), then the graph of f has a ______ (or oblique) ________

slant asymptote

do you put brackets around solutions that make the bottom 0

no

when you divide by a - in an inequality, ____ the sign

flip

in an inequality coordinate plane graph, the shaded area are the ______ and the line indicates ________

values that make the equation true, value that makes the equation =

3 key numbers means

4 test regions

key numbers are ____

whatever makes the top and bottom 0

are solutions in interval notation ordered pairs

no

what is the radicand

the polynomial under the root

special cases are also called ______

unusual solution sets

key values with solutions in the middle are called ____ points

border

between two consecutive zeros, a polynomial must be entirely ____ or entirely _____

positive, negative

To solve a polynomial inequality, find the _______numbers of the polynomial, and use these numbers to create __________ ________ for the inequality

critical/key, test intervals

the key numbers of a rational expression are its ______ and its ___ ____

zeros, undefined values

the formula that relates cost, revenue, and profit is ______

profit=revenue-cost

how can you tell if an equation has 4 test regions

find solutions and make sure there are 3 roots

Can you cross through a HA? VA? SA?

Yes no yes

All graphs _____ the asymptotes

Follow

Make sure graphs don’t cross if there aren’t enough X intercepts

..

What is root 3i times root 3i

3i^2

-3

Increasing and decreasing intervals use the ___ values

X (watch asymptotes)

Increasing and decreasing intervals use the ___ values

X (watch asymptotes)

When finding key points of rational functions ___ first

Simplify

Why is x^4 + x^2-60 guaranteed two real roots

It’s down 60 going up eternally

Can an answer written as the product of linear factors have imaginary factors?

Yes, they are still linear