PreCalc Chapter 2 Test Part 2 Flashcards
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
