1. THE LINEAR FACTORIZATION THEOREM

2. What is the Linear Factorization Theorem? If where n > 1 and a n ≠ 0 then Where c 1, c 2, … c n are complex numbers (possibly real and not necessarily distinct)

3. What Does the Linear Factorization Theorem Tell Us?  First, it tells us that the total number of zeroes of a polynomial, multiplied by their multiplicities, is the degree of the polynomial.  Second, it allows us to find a polynomial that has whatever zeroes we want.  We can accomplish this by multiplying together factors – one (or more) for each zero.

4. Building a Polynomial Using the Linear Factorization Theorem 1. Determine all the zeroes you want your polynomial to have and what multiplicity each should have. 2. Generate a factor for each zero. 3. Multiply together all the factors. Multiply by each one a number of times equal to its multiplicity.

5. Example  Build a 5 th degree polynomial that has roots 2, -1, and 1 + i.

6. Solution: Step 1  First, we remember that complex zeroes must come in conjugate pairs.  If we’re going to have 1 + i as a zero, we need to have 1 – i as well.  In addition, to get a 5 th degree polynomial, one factor will have to have multiplicity 2.  We’ll arbitrarily decide to give 2 multiplicity 2.

7. Solution: Step 2  With these zeroes, our factors are (x – 1 – i), (x – 1 + i), (x – 2), and (x + 1).

8. Solution: Step 3  Now, we’ll multiply our factors together.  The product will be (x – 1 – i)(x – 1 + i)(x – 2) 2 (x+1)  The (x – 2) term is squared because it has multiplicity 2.  We expand and simplify, giving us a final result of x 5 – 5x 4 + 8x 3 – 2x 2 – 8x + 8