Consider the polynomial
$p(x) = x^5 - 7x^4 + 20x^3 - 36x^2 + 27x - 5$. Using
some combination of Synthetic division, Long
Division, the Complex Conjugate Theorem, the
Rational Zero Test and the Factor Theorem, explain
and demonstrate how to do the following: a) Find all
the solutions (real and imaginary) of the polynomial
equation $p(x) = 0$. Note: $1 + 2i$ is a zero of $p(x)$. b)
Factor $p(x)$ completely over the complex number
field. Do not use technology for teaching Part 1
except to produce a good graph of the polynomial.
Finally, use the graph produced (bring the graph you
produced) by a graphing technology to explain the
relationship between the graph of the polynomial and
the solutions of the polynomial equation.
