Thread: Help... I'm stuck with a problem

I'm kinda stuck with two types of problems given to me. First is about division of algebraic expressions using long division and the other is synthetic division. I'm kinda confused with what i'm supposed to do next. Here are the problems:

Long division:
1. 12x^3-11x^2y-12xy-25y^3 / 3x-5y
2. 9m^3 - 32m^3 + 42m^6 - 15n^2 + 6 - n / 7m^3 + 5n - 3

The first two, I'm stuck since I couldn't find a similar term to subract the product I got.

3. 6xy - 6xz - 2yz + z^2 + y^2 + 9x^2 / 3x + y - z

I got 3x + 2y - z - (-y^2-yz/3x+y-z) though I'm not entirely sure with my answers.


Synthetic Division:
1. x^5-y^5 / x-y
2. (x^2-xy+y2) / (x+y)
3. (x^3 + x^2y+xy^2+y^3) / (x-y)

ok the first two. 3x goes into 12x^3 4x^2 times. 4x^2*(3x-5y)=12x^3-20x^2y


_ 4x^2-3xy+1______________
3x-5y )12x^3-11x^2y-12xy^2-5y^3
-(12x^3-20x^2y)
------------------
0 + 9x^2y-12xy^2
-(9x^2y-15xy^2)
-------------------------
0 + 3xy^2-5y^3
-(3xy^2-5y^3)
------------------
0
Synthetic division works like this:

. x^5-y^5 / x-y, think of this as 1*x^5+0*x^4+0*x^3+0*x^2+0*x^1+y^5*x^0/(1*x^1-y)

Ok this works when dividing by (x-a), in your case a=y

write "a" in a box on the left and all the coefficients of the polynomial on the right:

y] 1 0 0 0 0 y^5
y -y^2 y^3 -y^4 y^5
-----------------------------------------
1 -y y^2 -y^3 y^4 0
so, we "drop" the 1 and multiply by y. that goes in the next space where we subtract from zero and get -y^2 which we subtract from zero getting y^2. We continue in this fashion until we run out of terms.
We now use the numbers in the bottom to form a new polynomial of one less degree. (1)*x^4+(-y)*x^3+(y^2)*x^2+(-y^3)*x^1+(y^4)*x^0+0/(x-y)
That zero over x-y comes from the remainder term. If this hadn't divided evenly the last difference in the division would be nonzero and would get placed over x-y where that zero is. Our answer, though, is
x^4-y*x^3+y^2*x^2-y^3*x+y^4.
I must mention that it is critical that for any "missing powers" in your polynomial you put zeros in your synthetic division or else it WON'T WORK.
Hope this helps.