add x/x+2 + x^2 + 2x - 8

Question

This is basic math tripping me up, but i think my brain is fried.  I multiplied the last three terms by x+2 to get rid of the fraction.  Is that right?

Answer 1

     A a useful technique some times is to take an easier version of a similar problem. Do you know how to add 3/4 + 6? That's 3/4 + 6/1, then make common denominators by multiplying the second fraction by 4/4 resulting in 24/4. So you can now combine the numerators to get 27/4.

     On this problem, if you trid your method, you'd get 3+24 which is 27, an incorrect answer. I think I know where your confusion is though and I'll address it later.

    First, we'll do your problem correctly.

You need to multiply your last 3 terms by (x+2)/(x+2) which is 1. Making common denominators. you get

x/(x+2) + (x^3 + 2x^2)/(x+2) + (2x^2 + 4x)/(x+2) + (-8x-16)/(x+2).

 

Now that yo have common denominators, you can combine the numerators.

your answer as a single fraction is (x^3 + 4x^2 - 3x -16)/(x+2).

 

 

Now you may have been confusing this problem with a fractional (or rational) equation. For example,

4/(x+2) + 3/(x+3) = 5/(x+2).   Here the common denominator is (x+2)(x+3). To get rid of all the fractions, some teachers ask you to

                                                   multiply both sides of the equation to get rid of the denominators. This is allowed in equations since you did

                                                   the same thing to both sides. You cannot do that with an expression.

So multiplying by (x+2)(x+3) we can transform the equation into

4(x+3) + 3(x+2) = 5(x+3).

 

The expressions have changed but the equation is still in balance.

Hope this makes some sense to you.