How does Synthetic Division for linear divisors $ax + c$ with $a>1$ work?

Question

I used this guide from Mesa Community College to learn synthetic division. However it does not seem to work if $a>1$ in the divisor $ax + c$.

For example for this problem $\frac{3x^3-5x^2+4x+2}{3x+1}$ from the same website when I expand the solution in the picture below from the website I get $$(3x^2-6x+6) (3x+1) = 9x^3-15x^2+12x+6\neq 3x^3-5x^2+4x+2$$ So are they wrong? How can synthetic division be done correctly for $a>1$? I also noticed that the expanded solution can be divided by three to get the expected polynomial, how can this be integrated into the synthetic division algorithm?

Answer 1

Rewriting $$ \frac{3x^3-5x^2+4x+2}{3x+1}=\frac{3x^3-5x^2+4x+2}{3\left(x+\frac{1}{3}\right)} $$ you see that you have to divide your result, obtained not taking into account the facotor $3$, by $3$.