Pre-Calculus Group ProjectGroup Members:

Factoring Polynomials

In class, we discussed the Rational Roots Theorem which provides a complete list of possiblerationalrootsof thepolynomialequationanxn+an–1xn–1+ ··· +a2x2+a1x+a0= 0 where allcoefficientsareintegers.This list consists of all possible numbers of the formp/q, wherepandqare integers,pis a factor of theconstant terma0, and qis a factor of the leading coefficientan.

List all possible rational roots of

Sometimes our list of possible rational roots can be quite long and checking all of them can be time consuming before we find the first actual root. Fortunately, there are two other theorems that can cut down your work.

Upper Bound Theorem:
If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all non-negative numbers, then c is an upper bound to the real roots of the equation f(x) = 0.Note: Non-negative numbers include positive numbers and zero.
Lower Bound Theorem:
If you divide a polynomial function f(x) by (x - c), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f(x) = 0. Special note that zeros can be either positive or negative.

Use synthetic division to test if (x – 6)is a factor of

Notice that the division yields all non-negative numbers. That means 6 is an upper bound so we don’t need to check any of our possible rational roots higher than 6.

Use synthetic division to test if (x + 3)is a factor of

Notice that the division yields alternating signs. That means -3 is a lower bound so we don’t need to check any of our possible rational roots lower than -3.

Therefore, we only need to check values between -3 and 6. Lets try -1.

Since division by -1 yielded a remainder of 0, (x + 1) is a factor. Thus we can rewrite our polynomial as (x + 1)().

Let’s try another potential root. Keep working until you get only factors!

Special note: These theorems only work in one direction. Trying a potential root may not give you all non-negatives but it may still be true that no values above are roots.

Factoring Project

YOU MAY NOT USE A CALCULATOR FOR THIS PROJECT. You may only use it to do multiplication, division, subtraction and division. No graphing allowed.

For the polynomial

  1. Find all possible rational roots.
  2. Use synthetic division to factor the polynomial by hand. You will have to do this multiple times. Show ALL WORK.
  3. If you discover an upper bound, make note of it.
  4. If you discover a lower bound, make note of it.
  5. Sketch the graph using your knowledge of end behavior and multiplicity.

Format your work in an organized form to submit. All work and answers should be included. This can be done in a report, powerpoint presentation, video, website, etc.

You will be filling out a peer assessment to evaluate the work of yourself and your group.