2.  The remainder theorem states that the remainder that you get when you divide a polynomial P(x) by (x – a) is equal to P(a).  The factor theorem is a direct consequence of the remainder theorem.

3.  The factor theorem states that (x – a) is a factor of a polynomial P(x) if and only if P(a) = 0.  This is because if P(a) = 0, the remainder when P(x) is divided by (x – a) is zero as well.

4.  The factor theorem lets us easily factor polynomials.  If we can find a point a where P(a) = 0, we know that we’ve found a root of P(x) and that we can factor (x-a) out of P(x).  We can do this using synthetic division or long division.

5.  Find all the roots of f(x)=2x 3 + 3x 2 – 11x – 6.  First, we should find the possible rational roots of the function. Using the rational roots test, we find that they are ±1, ±2, ±3, ±6, ±1/2, and ±3/2.  Let’s start testing points to see if we can find a root.

6. xf(x) 1-12 6 20 f(2) = 0, so (x-2) is a factor of our polynomial. Let’s divide f(x) by x-2 using synthetic division to begin factoring it.

7.  This tells us that we can factor f(x) as (x – 2)(2x 2 + 7x + 3).  We can factor this quadratic comparatively easily. Our final result is (x – 2)(2x+1)(x+3).  If we didn’t have a quadratic, we could use synthetic division again on our quotient.  The roots of our function are 2, -1/2. and -3.