# 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 y-intercept, doing
simple sign -analysis, you should get a

graph similar to:

Figure 1: Partial graph of f (x) = x^{4} - 4x^{3} - 16x^{2} + 36x +
63.

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