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