Question

Apply division algorithm to find the quotient and remainder on dividing the polynomial 3 x^ + 4 x^ square + 6 x + 9 by x^ + 3x + 7 also verify the division algorithm

Answer 1

I suppose you mean

`(3x^3+4x^2+6x+9)/(x^2+3x+7)`

`3x^3=3x\cdot x^2`, and if we multiply
`x^2+3x+7` by
`3x` we get
`3x^3+9x^2+21x`. Subtracting this from the numerator gives a remainder of
`-5x^2-15x+9`.

`-5x^2=-5\cdot x^2`, and multiplying
`x^2+3x+7` by
`-5` gives
`-5x^2-15x-35`. Subtracting this from the previous remainder gives a new remainder of
`44`.

`44` has no remaining factors of
`x^2` in it, so we're done, and

`(3x^3+4x^2+6x+9)/(x^2+3x+7)=\underbrace{3x-5}_(\rm quotient)+\frac{\overbrace{44}^(\rm remainder)}{x^2+3x+7}`