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(4x^3+8x^2-x-2)/(x+2) =

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2 Answers

One method is "long division" . Using this method, you compare the leading terms of your numerator and denominator. 

How many times does x (from x + 2) go into 4x3 ? 4x2 times (x * 4x2 = 4x3). So we also multiply the 2 (from x + 2) by 4x2 (to get 8x2), and now we subtract it from our numerator, and we are left with (-x-2)/(x + 2)

Now, how many times does x go into -x ? -1 times. Multiply and subtract:

-x - 2 - (-1)(x + 2) = 0 

So our result is 4x2 - 1

__4x2 ______   -1  _________________

x + 2      | 4x3 + 8x2 - x - 2

4x3 + 8x2
---------------------

-x - 2
-x - 2

--------

      0

 

 

Comments

Synthetic Division is another method. 

In Synthetic Division, we "assume" that the denominator, x + 2, defines a zero; that is, x + 2 = 0 ==> x = -2

We then will set up our "division" problem, using only the coefficients and our "zero":

-2    |  4   8   -1   -2
       |____________

We bring down our first coefficient unchanged:

-2 | 4 8 -1 -2
    |____________
      4

Then we multiply our zero by the number we brought down, and place in our problem:

-2 | 4   8   -1   -2
    |___-8_________
       4   0

We repeat for each term:

-2 | 4    8   -1   -2
    |___-8__0  _______     <=== -2 * 0 = 0, so that gets inserted
      4    0    -1

-2 |   4   8   -1   -2
    |____-8__0 __2_____ 
        4    0   -1    0

So our result would be 4x2 - 1, which is the same result we got above. 

I'm not sure how synthetic division would work if you didn't have a linear divisior (denominator), such as the problem (4x4 + 8x3 + 2x2 - 8x - 8)/ (x2 - 3x + 2), but long division would work in this case. 

 

Comment

A way to check your answer, which is correct from Kevin S., is to multiply both the numerator and denominator by the original denominator (x+2).  As shown:

(4x^2 - 1) / 1 * (x + 2) / (x + 2)  (doing this step is effectively multiplying your answer by "1")

(4x^2 - 1)*(x + 2) / (x+2)

(4x^3 + 8x^2 - x - 2) / (x + 2)  (FOIL)

Since we got the same answer, the solution checks out.