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Material Type: Assignment; Class: Complex Variables W/ Apps; University: University of Hawaii at Hilo; Term: Unknown 1989;
Typology: Assignments
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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)
(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.