Synthetic Division Problem

[Alan51q]

Hello,

I've been working through a section on synthetic division, and in general have found it fairly comprehensible, having made few mistakes. That said, I'm having a hard time with the very last problem of the section. I've spent about an hour twisting it about, and can't figure out what I'm doing wrong. The answer, according to the book, is -9. However, I just can't figure out how to get there. Here is the problem:

Find the sum of a, b, c, and d if:

(x^3 - 2x^2 + 3x + 5)/(x+2) = ax^2 + bx + c + d/(x+2)

I figured since this is a synthetic division chapter that I should probably start there, so I used it on the left side of the problem to figure out, I think, that:

x^2 - 4x + 11 + 17/(x+2) = ax^2 + bx + c + d/(x+2)

I wasn't sure how to proceed from there, but tried multiplying both sides by (x+2) to get rid of the fractions:

(x+2)(x^2 - 4x + 11) + 17 = (x+2)(ax^2 + bx + c) + d

I'm not even sure that this is the right idea, or correct for that matter, and if it is, I still haven't been able to figure out how to remove x from the right of the problem to figure a+b+c+d. Clearly I'm missing something here, and any help in the right direction would be appreciated.

[anonmeans]

The answer, according to the book, is -9. However, I just can't figure out how to get there. Here is the problem:

Find the sum of a, b, c, and d if:

(x^3 - 2x^2 + 3x + 5)/(x+2) = ax^2 + bx + c + d/(x+2)

I figured since this is a synthetic division chapter that I should probably start there, so I used it on the left side of the problem to figure out, I think, that:

x^2 - 4x + 11 + 17/(x+2) = ax^2 + bx + c + d/(x+2)

Check your signs. I'm pretty sure the fraction should be subtracted instead of added.

I wasn't sure how to proceed from there...

Once you have "x^2 - 4x + 11 - 17/(x+2) = ax^2 + bx + c + d/(x+2)", do what they call "equating coefficients": the variables and fraction parts are all the same (after the division) so the coefficients have to be the same. So x^2 = 1x^2 = ax^2, so it has to be a = 1. Etc. Get all the coefficients. Then add them up.

[Alan51q]

Check your signs. I'm pretty sure the fraction should be subtracted instead of added.

Once you have "x^2 - 4x + 11 - 17/(x+2) = ax^2 + bx + c + d/(x+2)", do what they call "equating coefficients": the variables and fraction parts are all the same (after the division) so the coefficients have to be the same. So x^2 = 1x^2 = ax^2, so it has to be a = 1. Etc. Get all the coefficients. Then add them up.

It occurred to me that each side looked very similar, and so I did try to do as you suggest, but with the sign mistake I'd made in front of the fraction on the left, it didn't work of course. At least when I waste so much time trying to figure something out because of a mistake that could easily have been caught, it ingrains the idea in my head to go back and _double-check_.

Thanks again for the help!

BTW, do you know if graphing calculators are able to equate coefficients? I assume that they are. I tried entering the problem in my calculator (ti-89), to see if I could get -9, and it returned an error message of some sort. I believe that I chose Solve(). Then after the equation, put ",a+b+c+d"

I am still familiarizing myself with the calculator, so I'm sure that I just entered things incorrectly. For instance, I used the "Alpha" button to get a,b,c,d into the equation, and I'm not sure if that is allowed.

Anyway, thanks again.

[maggiemagnet]

"Solve" would be trying to solve an equation for the value(s) of the variable. I can't think of any utility that would do what you're asking.