polynomials
zeros & factors
inequalities
helpful theorems & rules
graphs
100
Find the polynomial with zeros: 2 multiplicity 2;3+2i write in factored form
f(x)=(x-2)^2(x-(3+2i))(x-(3-2i))
100
Find the real zeros and their multiplicities: f(x)=2(x-3)^2 (x+2)(x-4)
x=3 mult 2, -2,4
100
solve: (x+2)^2(x-1)(x+3)≥0
(-∞,-3]U{-2}U[1,∞)
100
This is used to find the possible number of positive and negative real zeros.
What is Descarte's Rule
100
what are the end behaviors for the graph of f(x)=-x^3+4x^2-3x+2
as x→-∞,f(x)→∞;as x→∞,f(x)→-∞ y=-x^3
200
If 6-5i is a zero for a polynomial with real coefficients, then name another zero.
6+5i
200
find the zeros of f(x)=x^3-3x^2-6x+8
-2,1,4
200
solve: x^3-x>0
(-1,0)U(1,∞)
200
This is used to find the least upper integral upper bound and greatest integral lower bound.
What is the upper/lower bound theorem?
200
describe the behaviors of the graph of f(x)=(x-4)(x+2)^2 (x-2) at each x intercept
both behaviors up, touches at -2, crosses at 2 and 4
300
If 2i is a zero for a complex polynomial, does it follow that -2i is a zero? explain
no. if a poly has complex coefficients, non-real zeros do not have to come in conjugate pairs.
300
Factor the polynomial so that all factors are linear. f(x)=4x^3-4x^2-7x-2
f(x)=4(x-2)(x+1/2)^2
300
solve: 3x^3<-15x^2
(-∞,-5)
300
This theorem states that every complex polynomial has at least one complex zero.
What is the fundamental theorem of algebra?
300
graph f(x)=-2x^3+4x^2
left behavior up, right down, cross at x=2, touch at x=0
400
True or False? A negative integer can be an upper bound for the zeros of a polynomial.
False
400
Find the real zeros: f(x)=x^4-10x^3+23x^2x
5+2rad2,5-2rad2,0 mult 2
400
solve: 2x^3+3x^2-17x-30<0
(-inf,-5/2)U(-2,3)
400
This theorem states that for a polynomial with real coefficients, complex zeros come in conjugate pairs.
What is the conjugate pairs theorem?
400
f(x)=x^3-7x-6
like y=x^3 zeros at -1,-2,3 crosses at each
500
Explain why the polynomial f(x)=2x^3-4x+5 must have at least one real zero.
any polynomial of odd degree must cross the x-axis at least once because of the end behavior (or the intermediate value theorem)
500
Find all zeros of f(x)=3(x)^4+3x^3-17x^2+x-6, then write in factored form.
f(x)=3(x-2)(x+3)(x+ √3/3 i)(x-√3/3 i)
500
solve x^3+4x^2≥x+4
[-4,-1]U[1,∞)
500
This theorem states that if a
What is the Intermediate Value Theorem?
500
f(x)=6x^4+11x^3-16x^2-11x+10
like y=x^4 crosses xaxis at -5/2,-1,2/3,1
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