How do you simplify (6x^2-x+6 )/( 2x-1 )?

1 Answer
May 17, 2015

Use synthetic division:

I will write it up a little verbosely to make the process as clear as possible. Essentially it's like doing long division.

First notice that 3x(2x - 1) = 6x^2-3x, which will allow us to separate the x^2 term:

(6x^2 - x + 6) - 3x(2x - 1)

= (6x^2 - x + 6) - (6x^2-3x)

= 6x^2 - x + 6 - 6x^2 + 3x

=2x + 6

So 3x makes a good multiplier, with remainder #(2x+6).

To match the leading term 2x in the remainder, the next multiplier we want is 1:

1xx(2x-1) = 2x-1

Then

(2x + 6) - (2x - 1) = 2x + 6 - 2x + 1 = 7

This gives 7 as our final remainder. If the remainder was zero then we would have a simple factorization.

Add the multipliers that we have found together to get the term (3x+1).

So to summarize where we have got to so far:

(6x^2 - x + 6) = (3x + 1)(2x - 1) + 7

Therefore

(6x^2 - x + 6)/(2x - 1) = (3x + 1) + 7/(2x - 1)

= 3x + 1 + 7/(2x - 1)