1) A polynomial P is given.
P(x) = x^4 + 49x^2
(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. If a root has multiplicity greater than one, only enter the root once.)
x =

(b) Factor P completely.
P(x) =

2) A polynomial P is given.
P(x) = x^3 â?' 10x^2 + 50x
(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. If a root has multiplicity greater than one, only enter the root once.)
x =

(b) Factor P completely.
P(x) =

3) A polynomial P is given.
P(x) = x^3 â?' 3x^2 + 4x â?' 12
(a) Factor P into linear and irreducible quadratic factors with real coefficients.
P(x) =

(b) Factor P completely into linear factors with complex coefficients.
P(x) =

4) A polynomial P is given.
P(x) = x^4 + 12x^2 â?' 64
(a) Factor P into linear and irreducible quadratic factors with real coefficients.
P(x) =

(b) Factor P completely into linear factors with complex coefficients.
P(x) =

5) Find a polynomial with integer coefficients that satisfies the given conditions.
P has degree 2 and zeros
4 + i and 4 â?' i.
P(x) =

6) Find a polynomial with integer coefficients that satisfies the given conditions.
Q has degree 3 and zeros 2,
2i, and â?'2i.
Q(x) =

Solution Preview

1) Factor out x^2:
P(x)=x^2(x^2+49)
The zeros of P(x) are x=0 with multiplicity 2 (bacuse x^2 can be written as (x-0)^2) and the zeros of x^2+49, which are x=7i, x=-7i.
So, P(x)=x^2(x-7i)(x+7i)

2)P(x) = x^3 â?' 10x^2 + 50x
Factor out x: P(x)=x(x^2-10x+50) and solve the equation x^2-10x+50=0
we get ...

Solution Summary

Factoring out is demonstrated. Complex zeros are factored out..

1. Form a polynomial f(x) with real coefficents having the given degree and zeros
Degree 5; Zeros: 2; -i; -7+i
Enter the polynomial f(x)=a(____) type expression using x as the variable.
2. Find a bound on the real zeros of the polynomial function.
F(x)=x^4+x^3-4x-6
Every real zero of f lies between -____and ____ (its not

1) Factor the polynomial completely.
P(x) = x^2 + 9
P(x) =
Find all its zeros. State the multiplicity of each zero. (If an answer does not exist, enter DNE.)
real zero x=______ with multiplicity______
complex zero x=_______(positive imaginary part) with multiplicity ____
co

1. Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
F(x)=x^5-x^4+7x^3-8x^2-16x+13; [1.3,1.7]
Find value of f (1.3) ____ (simplify)
Find value of f (1.7) ______ (simplify)
2. Information is given about the polynomial f(x) whose coefficients a

I am trying to factor the polynomial f(x) = 2x^3 - 5x^2 - 4x + 3. I think it is (x-3)(x+1)(x- 1/2). Am I right? (See work below.)
Once I factor f(x), how do I use that to find the answers to the following questions?
a) f(x) = 0
b) f(x+2) = 0
c) f(2x) = 0
This is the work that I used to factor f(x):

1. Solve the inequality algebraically
5x-3≥-2x2
The solution set is ____(interval notation)
2. Solve the following inequality
60x-64<60/x
3. List the potential rational zeros of the polynomial function. Do not attempt to find zeros.
F(x)=11x^4-7x^3+x^2-x+1
4. Solve the equation in the real number system.

1) Factor the polynomial and use the factored form to find the zeros.
P(x) = x^3 â?' x^2 â?' 72x
x = (smallest value)
x =
x = (largest value)
2) Factor the polynomial and use the factored form to find the zeros.
P(x) = x^4 â?' 3x^3 + 2x^2
x = (smallest value)
x =
x =
x = (largest value)

Please see the attachment
Using the Pythagorean theorem (see attached) and knowing the sinz and cosz are analytic functions. What can we say about (see attached)
Show that cos(A+B)=cosAcosB-sinAsinB using exponential expressions for sin and cos
What are the zeros of

For each polynomial function, find all zeros and their multiplicities:
1. f(x)=(x+1)^2(x-1)^3(x^2-10)
Find a polynomial function of degree 3 with real coefficients that satisfies the given conditions:
1. Zeros of 2, -3, and 5; f(3)=6
2. Zero of 4 having multiplicity 2 and zero of 2 having multiplicity 1; f(1)= -18

When you're given Log(f(z)) of any kind of f(z), is there a particular method to find the domain of analyticity of Log(f(z)), how can you find its branch cuts?
I'm stuck with the following Log(1-1/z^2), i don't understand how to prove that the branch cut is [-1,1]