THE LINEAR FACTORIZATION THEOREM

What is the Linear Factorization Theorem?If where n > 1 and an ≠ 0then Where c1, c2, … cn are complex numbers  

(possibly real and not necessarily distinct)

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.

Building a Polynomial Using the Linear Factorization Theorem1. 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.

Example Build a 5th degree polynomial that has

roots 2, -1, and 1 + i.

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 5th degree

polynomial, one factor will have to have multiplicity 2.

We’ll arbitrarily decide to give 2 multiplicity 2.

Solution: Step 2 With these zeroes, our factors are (x – 1

– i), (x – 1 + i), (x – 2), and (x + 1).

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 x5 – 5x4 + 8x3 – 2x2 – 8x + 8