What is the correct way to divide complex polynomials in two steps?

[iamjon.smith]

Homework Statement

(3y-2/y+3) - (3y+1)/(y²+6y-9)

Homework Equations

The Attempt at a Solution

Ok, I have attempted to solve in 2 steps, step 1: solve 3y-2/y+3 step 2: solve 3y+1/y²+6y-9 and then subtract the answers. This doesn't seem to work, as I get:

3y-2/y+3 = 3 remainder 11
and
3y+1/y²+6y-9 = 3y+1/(y-3)(y-3)
*y²+6y-9 simplified = (y-3)(y-3)

so final answer I got was 3r11-[(3y+1)/(y-3)(y-3)], which doesn't make sense. I think I am approaching the problem wrong, please provide some direction...

 

[HallsofIvy]

Actually, "dividing polynomials" is not relevant here. This is a problem in subtracting fractions- and you need to get a common denominator.
The denominator of the first fraction is y+ 3 and the denominator of the second fraction is $y^2+6y- 9$. That is NOT (y- 3)(y- 3). If you were to multiply y-3 by y- 3 you would get $y^2- 6y+ 9$- the signs are wrong. In fact, if we were to "complete the square" we would get $y^2+ 6y+ 9- 9- 9= (y+ 3)^2- 18= (y+3- 3\sqrt{2})(y+ 3+ 3\sqrt{2})$. In any case that has no factors in common with x+ 3 so you need to multiply the numerator and denominator of the first fraction by $x^2+ 6x- 9$ and numerator and denominator of the second fraction by x+ 3.

 

[mfb]

What do you mean with "solve"? There is no equation you could solve.
Simplify? Then I don't understand your approach. Writing the second denominator as product is a good start (but check the signs)

Where do remainders come from? Division in the complex numbers does not have or need that concept.

 

[Ray Vickson]

Homework Statement

(3y-2/y+3) - (3y+1)/(y²+6y-9)

Homework Equations

The Attempt at a Solution

Ok, I have attempted to solve in 2 steps, step 1: solve 3y-2/y+3 step 2: solve 3y+1/y²+6y-9 and then subtract the answers. This doesn't seem to work, as I get:

3y-2/y+3 = 3 remainder 11
and
3y+1/y²+6y-9 = 3y+1/(y-3)(y-3)
*y²+6y-9 simplified = (y-3)(y-3)

so final answer I got was 3r11-[(3y+1)/(y-3)(y-3)], which doesn't make sense. I think I am approaching the problem wrong, please provide some direction...

You write
$$\left( 3y - \frac{2}{y} +3 \right) - \frac{3y+1}{y^2 + 6y - 9}$$
Do you mean that, or did you mean
$$\frac{3y - 2}{y+3} -\frac{3y+1}{y^2 + 6y - 9} \:?$$
If you meant the latter, use parentheses, like this: (3y-2)/(y+3) - (3y+1)/(y^2 + 6y - 9).

Anyway, $y^2 + 6y - 9 \neq (y-3)^2$. Are you sure you have copied out the problem correctly? The version you wrote does NOT simplify at all, but just becomes messier as you do more work on it.

 

[iamjon.smith]

My apologies to everyone that has attempted this so far, I transcribed the problem wrong, here is the actual problem:

(3y-2)/(y+3) - (3y+1)/(y^2 + 6y + 9)

In the original problem, I placed the parenthesis wrong AND put the wrong sign in one place...I am very sorry and need help with this corrected problem...

in which simplifying (y^2+6y+9) = (y+3)^2

 

[iamjon.smith]

You write
$$\left( 3y - \frac{2}{y} +3 \right) - \frac{3y+1}{y^2 + 6y - 9}$$
Do you mean that, or did you mean
$$\frac{3y - 2}{y+3} -\frac{3y+1}{y^2 + 6y - 9} \:?$$
If you meant the latter, use parentheses, like this: (3y-2)/(y+3) - (3y+1)/(y^2 + 6y - 9).

Anyway, $y^2 + 6y - 9 \neq (y-3)^2$. Are you sure you have copied out the problem correctly? The version you wrote does NOT simplify at all, but just becomes messier as you do more work on it.

Thanks, you caught where I copied the problem wrong, and where I misused the parenthesis, it was the latter, with +9 instead of -9!

 

[iamjon.smith]

With the problem copied correctly and reworked, I came up with:

(3y-2)/(y+3) - (3y+1)/(y+3)^2

(y+3)^2 LCD

(y+3)(3y-2) = 3y^2-2y+9y-6 =3y^2+7y-6

so now the problem has been simplified to:

(3y^2+7y-6)/(y+3)^2 - (3y+1)/(y+3)^2

(3y^2 + 7y - 6 - 3y -1)/(y+3)^2

(3y^2 + 4y - 7)/(y+3)^2

(y-1)(3y+7)/(y+3)^2 :correct answer??

 

[Ray Vickson]

With the problem copied correctly and reworked, I came up with:

(3y-2)/(y+3) - (3y+1)/(y+3)^2

(y+3)^2 LCD

(y+3)(3y-2) = 3y^2-2y+9y-6 =3y^2+7y-6

so now the problem has been simplified to:

(3y^2+7y-6)/(y+3)^2 - (3y+1)/(y+3)^2

(3y^2 + 7y - 6 - 3y -1)/(y+3)^2

(3y^2 + 4y - 7)/(y+3)^2

(y-1)(3y+7)/(y+3)^2 :correct answer??

Yes.