A Strange Issue with Synthetic Division

[tawny sigil]

I'm trying to perform (\frac{2x^{3}-3x^{2}-8x-3}{2x+1}) to factor the numerator, and the complete table of Synthetic Division I end up with is:

[\begin{array}{c|rrrr}
-\frac{1}{2} & 2 & -3 & -8 & -3 \
& & -1 & 2 & 3 \
\hline
& 2 & -4 & -6 & 0
\end{array}]

But strangely, the answer (2x^{3}-3x^{2}-8x-3=(2x+1)\left(2x^{2}-4x-6\right)) appears to be wrong; the true answer is (2x^{3}-3x^{2}-8x-3=(2x+1)\left(x^{2}-2x-3\right))... Why?

[faint nymph]

Lober

[tawny sigil]

I've tried factoring the numerator by grouping before trying Synthetic Division to see if anything simplifies but if there is something to find to that end I cannot seem to figure it out

and strangely, long division doesn't run into this issue; is this some unnamed limitation of Synthetic Division?

[ruby light]

try long division instead

synthetic division is only used when the divisor is a linear equation of x-a, where the coefficient of x is only 1

in this case, long division or reverse area model works best

[tawny sigil]

I see a contradiction here

[ruby light]

ok i didnt read this part

[tawny sigil]

so is this paper of ours just blatantly false on this?

[ruby light]

i never solved polynomial division using synthetic divison when the coefficient is greater than 1. i've only ever used long division for convenience

so i cant say if that paper is wrong

[ruby light]

do you mean there's a remainder

[tawny sigil]

No I mean long division does give the correct answer where synthetic division doesn't

[ruby light]

Synthetic division can only be used if you're dividing by a LINEAR factor in the form x - a. For example, it works if you are dividing by x - 3 or by x + 4. If you need to divide by a polynomial with a higher degree or one that doesn't have a 1 as the leading coefficient, you'll have to do plain old long division.

https://www.katesmathlessons.com/synthetic-division.html#:~:text=Synthetic division can only be,do plain old long division.

[tawny sigil]

I see

[ruby light]

synthetic divison only works if the coefficient is 1

yeaaaa

[tawny sigil]

that makes it very useless lmao wow

okay well thanks that is a strange error

[ruby light]

lmao ur welcome

[tawny sigil]

.solved