how to do this, Divide (3x 4 + 19x 3 − 25x 2 − 57x + 150) by (−x − 7).

Answer 1

There are two ways of doing polynomial division. One method that always works it by using polynomial long division, and the other way is through synthetic division. Note that synthetic division only works when the divisor consists of a degree 1 polynomial (as is the case here). Both methods will be covered below:

Polynomial long division:

-3x^3 + 2x^2 + 11x - 20

-x-7 / 3x^4 + 19x^3 - 25x^2 - 57x + 150

- 3x^4 + 21x^3

_____________

-2x^3 - 25x^2 - 57x + 150

- -2x^3 - 14x^2

_______________________

-11x^2 - 57x + 150

- -11x^2 - 77x

___________________

20x + 150

- 20x + 140

_______________

10

Thus we have our remainder is 10 and thus we see that this method yields:

3x^4 + 19x^3 - 25x^2 - 57x + 150 / (-x-7) = -3x^3 + 2x^2 + 11x - 20 + (10/(-x-7))

For synthetic division, we first cancel out the negative sign on the divisor and write out

-7 | -3 -19 25 57 -150

| 21 -14 -77 140

_________________________

-3 2 11 -20 -10

and get that

3x^4 + 19x^3 - 25x^2 - 57x + 150 / (-x-7) = -3x^4 - 19x^3 + 25x^2 + 57x - 150 / (x+7) = -3x^3 + 2x^2 + 11x - 20 + (10/(-x-7))

which is exactly the answer we got from polynomial long division.