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Complex Variables - 4 Questions on Homework | MATH 303, Assignments of Mathematical Analysis

Material Type: Assignment; Class: Complex Variables W/ Apps; University: University of Hawaii at Hilo; Term: Unknown 1989;

Typology: Assignments

2009/2010

Uploaded on 04/12/2010

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Math 303 - Complex Variables
Homework due January 16
Question 1. In class, we used the quadratic equation to show that certain quadratic polynomials
cannot have any real roots. Recall that if our quadratic equation has the form ax2+bx +c, then
its roots are given by
x=b±b24ac
2a.
Our discussion centered on the the problems that manifest when we radicand b24ac is negative.
We did not, however, investigate the situation where the discriminant is exactly zero (and thus we
should get out a valid answer, since we can take the square root of zero).
(a) Consider the polynomial p(x) = x26x+ 9. Graph this polynomial. What are its roots?
(b) Use the quadratic equation to find the roots of this polynomial. Does it match your answer
from (a)?
(c) Factor the polynomial x26x+ 9 into simpler terms. Does this give the same roots as in (a)
and (b)?
(d) The above situation is known as a double root, since the root you found seems to appear with
multiplicity two. Construct a polynomial that has 2 as a double root.
(e) Construct a polynomial that has 2 as a double root and 3 as a “single” root. Graph it.
What looks graphically different about the double root versus the single root?
Question 2. We learned in class that iwas the root to the polynomial p(x) = x2+ 1. However, as
a quadratic, we expect to have two roots. What is the other root?
Question 3. Write the following complex numbers in the form a+bi.
(a) (1 + i)3
(b) 2 + i
6i(1 2i)2
(c) 3
i+i
3+8i1
i
Question 4. Find a general formula for ikfor any kZ. Be sure to include the case when kis
negative. Use your formula to compute i1827361.
1

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Math 303 - Complex Variables

Homework due January 16

Question 1. In class, we used the quadratic equation to show that certain quadratic polynomials cannot have any real roots. Recall that if our quadratic equation has the form ax^2 + bx + c, then its roots are given by

x =

−b ±

b^2 − 4 ac 2 a

Our discussion centered on the the problems that manifest when we radicand b^2 − 4 ac is negative. We did not, however, investigate the situation where the discriminant is exactly zero (and thus we should get out a valid answer, since we can take the square root of zero).

(a) Consider the polynomial p(x) = x^2 − 6 x + 9. Graph this polynomial. What are its roots?

(b) Use the quadratic equation to find the roots of this polynomial. Does it match your answer from (a)?

(c) Factor the polynomial x^2 − 6 x + 9 into simpler terms. Does this give the same roots as in (a) and (b)?

(d) The above situation is known as a double root, since the root you found seems to appear with multiplicity two. Construct a polynomial that has −2 as a double root.

(e) Construct a polynomial that has −2 as a double root and 3 as a “single” root. Graph it. What looks graphically different about the double root versus the single root?

Question 2. We learned in class that i was the root to the polynomial p(x) = x^2 + 1. However, as a quadratic, we expect to have two roots. What is the other root?

Question 3. Write the following complex numbers in the form a + bi.

(a) (1 + i)^3

(b)

[

2 + i 6 i − (1 − 2 i)

] 2

(c)

i

i 3

8 i − 1 i

Question 4. Find a general formula for ik^ for any k ∈ Z. Be sure to include the case when k is negative. Use your formula to compute i^1827361.