Question

How do you divide `(-x^4-8x^3-x^2+4x-2)/(x^2+4)`?

Answer 1

Quotient = -x² -8x +3
Remainder= 36x -14

Explanation:

Step 1] Write down the coefficients of the polynomial in the numerator.

Here, -1 -8 -1 +4 -2

[Note; if there's a missing degree in between, write the coeff. as 0]

Now, same with denominator
Coeff. +1 0 +4

Step 2] Now, after noting down the coeff. start with first coeff. of denominator. Using that, divide the numerator's first coeff.
Here, we get -1 on dividing.

Step 3] Now, the important step,
use this number (the one you got after dividing) and

multiply through with the coeffs. of the denominator (i.e. here, multiply -1 with all coeffs. of denominator) and

write this new set, just below the numerator, (starting from left)

Num. coeff. -1 -8 -1 +4 -2
New set -1 0 -4

Step 4] Now, in the next step, you just subtract those two sets, like this

     -1 -8 -1 +4 -2

(-) -1 0 -4
———————————————
0 -8 +3

Step 5] Now, step 2 repeats, not exactly same though.
Now the resultant numbers that we've got (here, -8 +3), they will act like the numerator.

Similarly, like in step 2, Divide the first coeff., by the first coeff. of the denominator.
We get -8.
And similarly again, multiply through with the denominator's coeffs.
And also, subtract it to get a new set again.
We have, [+4 and -2 have been just brought down]

     -8 +3 +4  -2

(-) -8 0 -32
————————————
0 +3 +36 -2

Again (like step 5), doing same
We get
+3 +36 -2
(-) +3 0 12
—————————
0 36 -14

Now, we've reached the end (as all coeffs. in the original numerator are done working with).
In doing so, we've collected important numbers we need (i.e. the quotient)
The numbers obtained when we divided numerator's (also the ones in further steps) first coeff. by denominator's first coeff.
Those numbers, here are -1 -8 +3
These numbers are actually the coeff. of our quotient. The x's are just to be inserted with correct degree.

Degree entry is simple, look at the degree of the original numerator (here 4, it's the highest power of x)
Divide this by the degree of denominator (here 2)
And we get 2 (this will be our degree of quotient, and subsequent powers will be one less)

Hence, the required quotient is
-1x² -8x + 3
With the remainder being 36x -14 (from the set obtained in last step, with correct indices for x)

Answer 2

`-x^2-8x+3 + (36x-14)/(x^2+4)`

Explanation:

Using place keepers of no value. For example `0x^3`

`color(white)("¬¬¬¬¬¬¬¬¬¬¬.")-x^4-8x^3-x^2+4x-2`
`color(magenta)(-x^2)(x^2+4) -> color(white)("d")ul( -x^4+0x^3-4x^2 larr" Subtract")`
`color(white)("¬¬¬¬¬¬¬¬¬¬¬¬¬¬") 0-8x^3 +3x^2+4x-2`
`color(magenta)(-8x)(x^2+4)->color(white)("¬¬¬¬.")ul(-8x^3+0x^2-32xlarr" Subtract")`
`color(white)("¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬")0+3x^2+36x-color(white)("¬")2`
`color(magenta)(+3)(x^2+4) ->color(white)("¬¬¬¬¬.¬¬¬¬¬")ul(3x^2+0xcolor(white)("¬")+12 larr" Subtract")`
`color(white)("¬¬¬¬¬¬¬..¬¬¬¬") color(magenta)("Remainder"->0 color(white)("¬")+36x-14 )`

`color(magenta)(-x^2-8x+3 + (36x-14)/(x^2+4)`