Question

What is `(8x^3 + 12x^2 - 6x + 8)/(2x+6)`?

Answer 1

`(8x^3 + 12x^2 - 6x + 8)/(2x+6) = (4x^2-6x+15) - 41/(x+3)`
or
quotient `= (4x^2-6x+15)` & remainder `= -82`

Explanation:

`(8x^3 + 12x^2 - 6x + 8)/(2x+6)`

dividing numerator and denominator by 2 (purely for simplification purpose)

= `(4x^3 + 6x^2 - 3x + 4)/(x+3)`

try to break the polynomial in the numerator such that we can factor out the denominator from successive terms

= `(4x^3 + 12x^2 - 6x^2 - 18x + 15x + 45 - 41)/(x+3)`2

=`(4x^2(x+3) - 6x(x+3) +15(x+3) - 41)/(x+3)`

=`(4x^2(x+3) - 6x(x+3) +15(x+3))/(x+3) - 41/(x+3)`

= `((x+3)(4x^2 - 6x + 15))/(x+3) - 41/(x+3)`

=`(4x^2-6x+15) - 41/(x+3)`

the second term in the expression `i.e. - 41/(x+3)` cannot be simplified further as degree of numerator is less than that of denominator.
[Degree is the highest power of variable in a polynomial
(`therefore` degree of `-41` is `0` and that of `x+3` is `1` ) ]

`therefore` `(8x^3 + 12x^2 - 6x + 8)/(2x+6) = (4x^2-6x+15) - 41/(x+3)`
or
quotient `= (4x^2-6x+15)` & remainder `= 2*(-41) = -82` since we had initially divided both numerator and denominator by 2, we have to multiply -41 by 2 to get the correct remainder.

Answer 2

`color(blue)(4x^2-6x+15`

Explanation:

`color(white)(aaaaaaaaaa)` `4x^2-6x+15`
`color(white)(aaaaaaaaaa)` `-----`
`color(white)(aaaa)2x+6` `|` `8x^3+12x^2-6x+8` `color(white) (aaaa)` `∣` `color(blue)(4x^2-6x+15)`
`color(white)(aaaaaaaaaaa)` `8x^3+24x^2` `color(white)`
`color(white)(aaaaaaaaaaa)` `----`
`color(white)(aaaaaaaaaaaaa)` `0-12x^2-6x`
`color(white)(aaaaaaaaaaaaaaa)` `-12x^2-36x`
`color(white)(aaaaaaaaaaaaaaaa)` `------`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaa)` `30x+8`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaa)` `30x+90`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaaa)` `---`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaa)` `-82`

The remainder is `=color(blue)(-82` and the quotient is `=color(blue)(4x^2-6x+15`