Question

If the quotient of two polynomials is `4x^2-x-7+(11x+15)/(x^2+x+2)`, what are the two polynomials?

Answer 1

`(4x^4 +3x^3+0x^2+2x+1) -: (x^2+x+2)`

Explanation:

Given:`" "4x^2-x-7+(11x+15)/(x^2+x+2)`

The `11x+15` is the remainder so the denominator must be the divisor. Thus we need to build each term by using `x^2+x+2`

Note that I use a place holder. For example: `0x^4`

color(white)(.)

`4x^2(x^2+x+2) -> 4x^4+4x^3+8x^2`
`-x(x^2+x+2)->0x^4-color(white)(4)x^3-color(white)(8)x^2-2x`
`-7(x^2+x+2)->0x^4+0x^3-7x^2-7x-14`
`"The remainder"->ul(color(white)(.)0x^4+0x^3+0x^2+11x+15) larr" Add"`
`" "4x^4 +3x^3+0x^2+2x+1`