Try our Free Online Math Solver!

Division of Mathematics and Physics Project #3
This project is due 03/12/2007. Be neat, orderly , and
answer every question. Answers without
work will not receive credit.
1. A polynomial equation is given along with information about factors. Use the
information
to solve the following equation.
x^{4} + 5x^{3}  23x^{2}  87x + 140 = 0
Given that x^{2} + x  20 is a factor of x^{4} + 5x^{3}  23x^{2}  87x + 140.
Work: First you'll need to use long division, and you'll find.
So we now have:
Clearly we have x = 5 and x = 4 as solutions, but we need
to use the quadratic formula
to find the zeros of x ^{2} + 4x  7.
So the solution set is:
2. Find a polynomial function , of smallest degree, with
integer coefficients , that has the
following as zeros: .
Work: Using these four roots we have:
So any polynomial, where k ≠ 0 ∈ Z, of the form
is an acceptable answer. For example, if k = 1, your answer is:
3. Given that
find:
Work:
Work:
(c) Use the above two facts to find all five roots of f .
Work: Your prior work with the quadratic formula should convince you that if
is a root then  is also a root; likewise if
is a root then 
is also a root.
We have one more to find.
So the remaining root is 1/2.
4. Use the Rational Root Theorem to factor and graph the
following function.
f (x) = x^{4}  4x^{3}  16x^{2} + 36x + 63
Work: Since it does not ask to list the candidate rational roots, I just want to nd two
reasonable ones that work.
f (3) = 0
f (3) = 0
Now using these two roots you should realize that
(x  3) (x + 3) = x^{2}  9
is a factor of f (x), and using long division we get
So, we have
The first factor's zeros are:
and the completely factor form of f is:
If you're graphing by hand you'll need to approximate.
and
After listing all x intercepts , the yintercept, doing
simple signanalysis, you should get a
graph similar to:
Figure 1: Partial graph of f (x) = x^{4}  4x^{3}  16x^{2} + 36x + 63.
Prev  Next 