Below is a series of expressions in the form \(\dfrac{p(x)}{d(x)}\), where \(p(x)\) and \(d(x)\) are polynomials.

What degree are the quotient and remainder polynomials when \(p(x)\) is divided by \(d(x)\)?

A. \(\dfrac{3x+1}{x}\)

B. \(\dfrac{5x-2}{2x-1}\)

C. \(\dfrac{3x^2-5x+2}{x+1}\)

D. \(\dfrac{x^3-5x^2+3x+1}{x+1}\)

E. \(\dfrac{5x^2+8x+9}{x^2+x+2}\)

F. \(\dfrac{x^3-x^2-7x+1}{x^2+2x-1}\)

G. \(\dfrac{x^4-3x^3 + x + 1}{x-1}\)

H. \(\dfrac{6x^4 + 5x^3 + x^2 +10 x -3}{2x^2 + 3x - 1}\)

Some of the questions have divisors which are quadratic. Below are suggestions as to how methods used for algebraic division by a linear divisor can be extended to working with a quadratic.

E. \(\dfrac{5x^2+8x+9}{x^2+x+2}\)

We could rewrite \(p(x)\) so multiples of the divisor, \(d(x)\), appear.

\[\dfrac{5x^2+8x+9}{x^2+x+2} = \dfrac{5x^2+5x+10+3x-1}{x^2+x+2}\]

Why is this helpful? What might the next step be?

F. \(\dfrac{x^3-x^2-7x+1}{x^2+2x-1}\)

We could use the grid method by extending the number of rows in the grid. For more support with using the grid method you may wish to look at Divide it up.

\(x\) | \(\ldots\) | |
---|---|---|

\(x^2\) | \(x^3\) | |

\(+2x\) | ||

\(-1\) |

Which cells would you fill in next?

We could also use long division:

\[ \begin{array}{rll} \phantom{x^2-x^2-7}x+\ldots & \cmlongdivbr x^2+2x-1 \cmlongdiv{x^3-x^2-7x+1} \cmlongdivbr \underline{x^3+\ldots\phantom{{}+1x+8}} && [x \times (x^2+2x-1)] \cmlongdivbr \end{array} \]