# How do you divide #(x^4 - 7x^3 + 2x^2 + 9x)/(x^3-x^2+2x+1)#?

##### 1 Answer

with the remainder

#### Explanation:

*I'm aware that in some countries, polynomial long division is written in a different format. I will use the format that I'm most used to, and I hope that it will be no problem for you to convert this into your prefered format.*

As you have asked how to do the division, let me explain the basic steps of polynomial long division:

1)Divide the highest power of your numerator by the highest power of the denominator.In your case, that's

#x^4 -: x^3 = x#

2)Multiply the result from 1) with your denominator.In your case, that's

# x * (x^3 - x^2 + 2x + 1) = x^4 - x^3 + 2x^2 + x#

3)Subtract the result from 2) from the numerator.In your case, that's

#(x^4 - 7x^3 + 2x^2 + 9x) - (x^4 - x^3 + 2x^2 + x) = -6x^3 + 8x#

4)Consider the result from 3) to be your new numerator.

Repeat the steps 1) - 3) as long as the highest power of your numerator is greater or equal to the highest power of your denominator.

Here's the total calculation:

# color(white)(xii) (x^4 - 7x^3 + 2x^2 + 9x) -: (x^3 - x^2 + 2x + 1) = x - 6#

# -(x^4 - x^3 color(white)(x)+ 2x^2 + x)#

# color(white)(x) color(white)(xxxxxxxxxxxxx)/ #

# color(white)(xxx) -6x^3 color(white)(xxxxx) + 8x#

# color(white)(x) -(-6x^3 + 6x^2- 12x - 6)#

# color(white)(xxx) color(white)(xxxxxxxxxxxxxxxx)/ #

# color(white)(xxxxxxxx) -6x^2 + 20x + 6#

Thus,

#(x^4 - 7x^3 + 2x^2 + 9x) -: (x^3 - x^2 + 2x + 1) = x - 6#

with the remainder

Or, if you prefer a different notation,

#(x^4 - 7x^3 + 2x^2 + 9x) -: (x^3 - x^2 + 2x + 1) = x - 6 + (-6x^2 + 20x + 6)/(x^3 - x^2 + 2x + 1)#