Lesson 3: Factor Theorem

What are Factors?

• Numbers or expressions that divide evenly into a dividend.
• This means that the modulus is 0. (remember the previous lesson?)

Factor Theorem

is a factor if and only if

• This is actually just a special case of the Remainder theorem.

Finding Factors

Which of the following are factors of

• ?
• ?
• ?
• ?
• ?

Finding Factors, cont.

Finding Factors, cont.

We can also find them using synthetic division.

Then, we get:

Integral Zero Theorem

Picking numbers is inefficient, so we need some way to make it efficient.

if is a factor of where is the constant term.

Finding Integral Roots

Find all possible integral roots of

1. Find the integral roots.
   • is prime, so it's only .
2. Synthetic Divide to a simpler form

Finding Integral Roots, cont.

We now have .

3. Factor if possible.

• $(x - 3)(x + 1)

Leaving us with:

All Possible Zeros

Find the possible zeros of

Here's our possible options:

Possible Zeros, Part 2

Here's our possible options:

1. Remove unreasonable answers

Remaining Possibilities: .

Possible Zeros, Part 3

2. Synthetic divide to simplify.

We get:

Possible Zeros, Part 4

We have:

3. Simplify any non-trivial polynomials

would be what's known as a bouncing root, but we'll get there when we do.

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