4x^4+3x^3+2x+1/x^2+x+2

Question

i need to divide this but i dont know how to factor the top part

Answer 1

Note that 4+1-(2+3)=0, so if x=-1, 4x⁴+3x³+2x+1=4-3-2+1=0, and x+1 must be a factor. Also note that there is no x² term and the denominator (divisor) cannot be factorised.

Using synthetic division we can find the other factor of the dividend (numerator):

-1 | 4  3 0  2    1 (0 is required for the missing x² term)

      4 -4 1 -1 | -1

      4 -1 1  1 |  0 = 4x³-x²+x+1.

There are no common factors between the denominator and numerator. [However, if the divisor had been x²-x-2=(x+1)(x-2), there would have been the common factor x+1.]

Using algebraic long division we get:

             4x    -5

x²+x+2)4x³    -x²  +x+1

             4x³+4x²+8x

                    -5x²-7x+1

                    -5x²-5x-10

                            -2x+11 remainder

Therefore (4x⁴+3x³+2x+1)/(x²+x+2)=(x+1)(4x³-x²+x+1)/(x²+x+2)=

(x+1)[4x-5 rem -2x+11].