(3x^2 + 11x + 1) / (x+ 4)

Question

Polynomial-long division

 

Answer 1

This is a long division since 3x²+11x+1 cannot be factored, therfore set up long division as follows:

                                    3x -1

                       X+4  Γ   3x²+11x+1  (1)

                                   (3x²+12x)    (2)   Subtract (2) from (1)

                                          -x +1   (3)

                                        (-x - 4)  (4)  Subtract (4) from (3)

                                               5          where 5 is the remainder

The answer is  3x -1 and the remainder is  5. This can be written as (3x-1) + (5 / (x+4))

the problem statement can be written as follows:

( 3x²+11x+1) / (x+4) = 3x-1 +(5/(x+4))

Proof:

Multiply (x+4) by ((3x-1)+(5/(x+4))) you get (x+4)(3x-1) + (5(x+4)/(x+4))

Expand you get 3x² -x + 12x -4 + 5 = 3x² +11x + 1 which is the numerator of the original expression

 

Answer 2

In order to do these kinds of problems, you want to generally factor the numerator polynomial. In this case since the numerator cannot be factored, you have to use long division. So the set up would be like this:

x+4 (long division sign here) (3x^2+11x+1)

we then realize that in order to get to 3x^2 from x+4, the closest thing we can do is multiply by 3x.

thus doing so, we would get 3x^2+12

we then subtract this from our original polynomial (3x^2+11x+1)

when we do that we get -x+1.

then we apply the same process try to multiply something with x+4 in order to get what we have under the division symbol. in this case we multiply by -1. We then get -x-4.

once again we subtract and we get 5. 

therefore our answer would be 3x-1 R. 5

when dealing with polynomial long divison we often show the remainder as part of the original fraction however thus making the final answer

 

3x-1 + (5)/(x+4)