High School Mathematics Extensions/Supplementary/Polynomial Division

Introduction
First of all, we need to incorporate some notions about a much more fundamental concept: factoring.

We can factor numbers,
 * $$5 \times 7 = 35$$

or even expressions involving variables (polynomials),
 * $$(x-3)(x+7) = x^2 + 4x - 21$$

Factoring is the process of splitting an expression into a product of simpler expressions. It's a technique we'll be using a lot when working with polynomials.

Dividing polynomials
There are some cases where dividing polynomials may come as an easy task to do, for instance:
 * $$\frac{x^3 + 6x - 12}{2x}$$

Distributing,
 * $$\frac{x^3}{2x} + \frac{6x}{2x} - \frac{12}{2x}$$

Finally,
 * $$\frac{1}{2}x^2 + 3 - \frac{6}{x}$$

Another trickier example making use of factors:
 * $$\frac{2x^3 + 3x^2 + 6x + 9}{2x + 3}$$

Reordering,
 * $$\frac{2x^3 + 6x + 3x^2 + 9}{2x + 3}$$

Factoring,
 * $$\frac{2x(x^2 + 3) + 3(x^2 + 3)}{2x + 3}$$

One more time,
 * $$\frac{(2x + 3)(x^2 + 3)}{2x + 3}$$

Yielding,
 * $$x^2 + 3$$

1. Try dividing $$35x^2 + 29x + 6$$ by $$2.5x + 1$$.

2. Now, can you factor $$P(x) = 3x^3 - 9x + 6$$ ?

Long division
What about a non-divisible polynomials? Like these ones:
 * $$(3x^2 + 3x - 4) / (x - 4)$$

Sometimes, we'll have to deal with complex divisions, involving large or non-divisible polynomials. In these cases, we can use the long division method to obtain a quotient, and a remainder:
 * $$P(x) = Q(x) \times C(x) + R$$

In this case:
 * $$(3x^2 + 3x - 4) = Q(x) \times (x - 4) + R$$

So finally:
 * $$(3x^2 + 3x - 4) = (3x + 15) \times (x - 4) + 56$$

3. Find some $$G(x)$$ such that $$(6x^2 - 13x + 7) - G(x)$$ is divisible by $$(3x + 1)$$.