Question

How do you divide using synthetic division: `(y^4 - 4y^2 + y + 4)/(y + 2)`?

Answer 1

`(y^4-4y^2+y+4)/(y+2) = (y^3-2y^2+1` with remainder `(+2)`

Explanation:

Synthetic Division

`{: (,,,color(brown)(y^4),color(brown)(y^3),color(brown)(y^2),color(brown)(y^1),color(brown)(y^0)), (color(brown)("[1] dividend coefficients"),,,1,(+0),-4,+1,+1), (color(brown)("[2]"),,,,-2,+4,+0,-2), (,,,"-----","-----","-----","-----","-----"), (color(brown)("[3] negative of constant from divisor"),xx(-2),color(blue)("||"),1,-2,0,1,color(red)(+2)), (,,,color(brown)(y^3),color(brown)(y^2),color(brown)(y^1),color(brown)(y^0),color(brown)("Remainder")) :}`

The test in `color(brown)("brown")` would not normally be written. It is there to help with the explanation only.

Line `color(brown)("[3]")`
- the value to the left of `color(blue)("||")` (when dividing by a monic binomial) is the negative of the constant term. In this case the constant term of the monic binomial (`x-1`) is `(-1)` so its negativeis `(+1)`
-the values to the right of #color(blue)("||") are the sum of the numbers in the column above each location. These values become the coefficients of the reduced polynomial terms, except for the final number which is the remainder.

Line `color(brown)("[2]")`
-is the product of the negative divisor coefficient (to the left of the `color(blue)("||")` in line `color(brown)("[3]")` and the sum from the previous column in line `color(brown)("[3]")` (to the right of the `color(blue)("||")`
:}#

The test in `color(brown)("brown")` would not normally be written. It is there to help with the explanation only.

Line `color(brown)("[3]")`
- the value to the left of `color(blue)("||")` (when dividing by a monic binomial) is the negative of the constant term. In this case the constant term of the monic binomial (`x-1`) is `(-1)` so its negative is `(+1)`
-the values to the right of #color(blue)("||") are the sum of the numbers in the column above each location. These values become the coefficients of the reduced polynomial terms, except for the final number which is the remainder.

Line `color(brown)("[2]")`
-is the product of the negative divisor coefficient (to the left of the `color(blue)("||")` in line `color(brown)("[3]")` and the sum from the previous column in line `color(brown)("[3]")` (to the right of the `color(blue)("||")`