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.

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.

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.

Find all the roots of f(x)=2x3 + 3x2 – 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.

x f(x)

1 -12

-1 6

2 0

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.

This tells us that we can factor f(x) as (x – 2)(2x2 + 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.