I'm not sure what to make of your question "why doesn't it have this term".

Do you mean why is the pattern different from what you expect (in which case, why do you expect your pattern to be true)? Or why is it -- that one statement -- true?

It is true because factoring is the opposite of multiplication -- X factors into Y*Z if and only if Z*Y multiplies out to X. And clearly this works in this case.

Now, note that you can do long division of your terms without any patterns.

t^2 -4t +16 dividing into t^3 + 0 t^2 + 0 t + 64

Take the highest term -- the t^2 -- on the left, and the highest term -- t^3 -- on the right. Well, t * (t^2) = t^3, so we start with that:

[t^3 + 0 t^2 + 0 t + 64] - [t^2 -4t +16] *

t= [t^3 + 0 t^2 + 0 t + 64] - [t^3 -4t^2 +16t]

= [4t^2 -16t +64]

Now do the same by dividing t^2 -4t +16 into 4t^2 -16t +64. Look at the highest coefficient -- t^2 and 4t^2. 4*t^2 = 4t^2.

[4t^2 -16t +64] - [t^2 - 4t +16] *

4= [4t^2 -16t +64] - [4t^2 -16t +64]

= 0

Now why did we do this? Because we are doing long division. Remember long division?

[t^3 + 0 t^2 + 0 t + 64] - [t^2 -4t +16] *

t =

[4t^2 -16t +64][4t^2 -16t +64] - [t^2 - 4t +16] *

4 = 0

Perform the italic substitution and get this:

[t^3 + 0 t^2 + 0 t + 64] - [t^2 -4t +16] *

t - [t^2 - 4t +16] *

4 = 0

Now, group the [t^2 -4t +16] terms:

[t^3 + 0 t^2 + 0 t + 64] - [t^2 -4t +16] * (

t + 4) = 0

Turn it into an equality:

[t^3 + 0 t^2 + 0 t + 64] = [t^2 -4t +16] * (

t + 4)

Now divide by t^2 -4t +16:

[t^3 + 0 t^2 + 0 t + 64]/ [t^2 -4t +16] = (

t + 4)

Drop the zero terms:

[t^3 + 64]/ [t^2 -4t +16] = (

t + 4)

See how that works?

What I just did there was long division, but I did it

backwards. As it happens, in this case, the remainder was 0, so the division was exact.

If you have (x^3 + 7) / (x+1), we can do the same thing:

x^3 + 7 -

x^2(x+1) = -x^2 + 7

-x^2 + 7 +

x(x+1) = x + 7

x+7 -

1(x+1) = 6

Each time, I am taking the remainder of the previous step, and dividing it again by the denominator.

We then get this:

(x+1)(x^2 - x + 1) = (x^3 + 7) - 6

which we can then check (my multiplying) to make sure we didn't do a mistake.

(x^2 - x + 1) = (x^3 + 7)/(x+1) - 6/(x+1)

In this case, we ended up with a remainder of 6. Had we ended up with a remainder of 0, it would have looked like the first case. Does this make sense?