I have an example of polynomial long division below. The sixth line down says $32x + 10$. I thought it would say 16x instead of 32x. I think I understand why: line five is "$-12x^2 -24x$." Since this is subtracted from the line above it ($-12x^2 +8x + 10$), "$-12x^2 -24x$" changes signs and becomes "$12x^2 +24x$." So is minus a minus ($-(-24x)$) the reason why the difference/subtraction on line six is 32x instead of +8x minus 24x? The line $-12x^2 -24x$, when subtracted becomes $12x +24x$, is that right? Here is the problem as transcribed from Carolyn Wheater's McGraw Hill 2010 "Practise Makes Perfect - Algebra" book:
$$ \require{enclose} \begin{array}{r} 3x^2 - 6x + 16 \\[-3pt] 2x + 4 \enclose{longdiv}{6x^3 + 0x^2 + 8x + 10} \\[-3pt] \begin{align} \underline{6x3 +12x^2}\phantom{2} \\[-3pt] -12x^2 + 8x + 10 \\[-3pt] \underline{-12x^2 - 24x}\phantom{2} \\[-3pt] 32x + 10 \\[-3pt] \underline{32x + 64}\phantom{2} \\[-3pt] -54 \\[-3pt] {} \end{align} \end{array} $$
Also, the MathJax tutorial and people who posted (esp. David K and David Cervone) was very helpful. But I could not figure out how to line up the various steps/levels of arithmetic both under the long-division sign and to the right in the comments. How do I line up the $x^2$'s under the $x^2$'s, the x's under the other x's, etc. Also I had wanted to include the Explanations (side comments) from the book I got this from on the side. I can do that using hline and some other examples I found, but it never lined up neatly for me. Below in the lefthand column are the expressions from the long division above, in the righthand column are Wheater's commentaries that I had wanted to include above but they made the unlined up example above more garbled. How do I make the two columns line up neatly? If MathJax specific questions go somewhere else, please let me know where.
$$ \require{enclose} \begin{array}{r} 3x^2 - 6x + 16 \\[-3pt] 2x + 4 \enclose{longdiv}{6x^3 + 0x^2 + 8x + 10} \\[-3pt] \begin{align} \underline{6x3 +12x^2}\phantom{2} \\[-3pt] \underline{-12x^2 +8x + 10} & \hbox{(-12x^2 \div 2x = -6x)} \\[-5pt] 4 | 500 \\ \underline{-12x2 -24\phantom{00}} & \hbox{($-6x times 2x +4$)} \\ 32x +10\phantom{0} & \hbox{($32x \div 2x = 16$)} \\ \underline{\phantom{0}32x +64\phantom{0}} & \hbox{($16 times 2x + 4$)} -54 \\[-3pt] {} \end{align} \end{array} $$
thanks,
Al
I think you understand polynomial division pretty well. Just make sure everything is neat and aligned, otherwise it is a nightmare to salvage. Below is the division you have posed rewritten:
$$ \require{enclose} \begin{array}{r} 3x^2-6x+16\\[-3pt] 2x+4 \enclose{longdiv}{6x^3+0x^2+8x+10} \\[-3pt] -\underline{(6x^3+12x^2)\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}~}\\ -12x^2+8x+10 \\ \underline{-(-12x^2-24x)\phantom{2}\phantom{2}~~} \\ 32x+10\\ \underline{\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}\phantom{2}-(32x+64)} \\ -54 \end{array} $$
I tried to align the numbers using \phantom{2} and ~. But in the font the $x^2$ is larger than a $x$, so it is much harder to align perfectly. I tried adding comments in the long division by using \tag{...} but it recognises the array as a whole object, so you can't choose which line to put the tag. Hope this helps somewhat.