How do you divide #(8x^3+5x^2-12x+10)/(x^2-3)#?

1 Answer
May 13, 2016

#(8x^3+5x^2-12x+10)/(x^2-3) = 8x+5 + (12x+25)/(x^2-3)#

Explanation:

I like to long divide just the coefficients, not forgetting to include #0#'s for any missing powers of #x#. In our current example that means the missing term in #x# in the divisor...

enter image source here

The process is similar to long division of numbers.

Write the dividend under the bar and the divisor to the left.

Write the first term #color(blue)(8)# of the quotient above the bar, choosing it so that when multiplied by the divisor #1, 0, -3# the product matches the first term of the dividend.

Write the product #8, 0, -24# of #8# and the divisor under the dividend and subtract it to give a remainder #5, 12#.

Bring down the next term from the dividend alongside it, then choose the next term #color(blue)(5)# of the quotient to match the leading term of our running remainder.

Write the product #5, 0, -15# under the running remainder and subtract it to give the final remainder #12, 25#.

There are no more terms to bring down from the dividend and the remainder is now shorter than the divisor, so this is where we stop.

Our quotient is #8, 5#, meaning #8x+5# and our final remainder is #12, 25# meaning #12x+25#

So:

#(8x^3+5x^2-12x+10)/(x^2-3) = 8x+5 + (12x+25)/(x^2-3)#