Math 3118, section 3
Spring 2001
Miscellaneous group problems for Tuesday, April 3
(Corrected version)
- Do the following calculations with complex numbers:
- Let a, b, c, and d be real numbers. Set z = a + bi and w = c + di.
Calculate
- =
- =
(Verify that the two expressions are equal.)
- If z = a + bi is a complex number, calculate the following:
- z2 = and
How are the answers related? (An answer of a few words should be possible!)
- z3 and
How are the answers related? (An answer of a few words should be possible!)
- zk
= and
Don't try to work it out algebraically; just try to figure out how the
two answers are related!
Let f(x) = x3 + 3x2 + 1, and let z = a + bi be a complex number.
How are the complex numbers f(z) and
f()
related to each other?
(Please note the correction here.)
Suggestion: The answers to parts a and b of the
previous problem can provide useful information.
The graph of f(x) = x3 + 3x2 + 1 is shown below:
- Use the graph to determine how many real solutions the equation f(x)= 0 has.
- Suppose that z = a + bi is a (non-real) solution of the equation f(z)= 0.
What is another solution of this equation?
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