\[\text{The answer to "$27$ divided by $4$" could be written as "$6$, remainder $3$"}\]

where \(4\) is the divisor, \(6\) is the quotient, and \(3\) is the remainder.

What other ways can you find to express \(\text{"$27$ divided by $4$"}\)?

There are many different ways we could write \(\text{"$27$ divided by $4$"}\). For example, \(6\dfrac{3}{4}\), \(\dfrac{24+3}{4}\) and \(6 + \dfrac{3}{4}\).

Why will the remainder will always be less than the divisor? Is a similar idea true in polynomial division? For example when a cubic polynomial is divided by a quadratic, what form will the remainder take?

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 \(d(x)\)?

After three questions we might already have some ideas as to the structure that is appearing. It may help to think about the following questions:

How do the questions differ? Does this explain the differences in the answers?

Can we make any conjectures about the outcome of the next question?

Can we always know what degree polynomial the quotient and the remainder will be by looking at the question?

The answers from above are summarised in the table.

Question | Degrees of \(p(x)\) and \(d(x)\) | Degree of \(q(x)\) | Degree of \(r(x)\) |
---|---|---|---|

A. | \(1, 1\) | \(0\) | \(0\) |

B. | \(1,1\) | \(0\) | \(0\) |

C. | \(2,1\) | \(1\) | \(0\) |

D. | \(3,1\) | \(2\) | \(0\) |

E. | \(2,2\) | \(0\) | \(1\) |

F. | \(3,2\) | \(1\) | \(0\) |

G. | \(4,1\) | \(3\) | Remainder of \(0\) |

H. | \(4,2\) | \(2\) | \(1\) |

\(m,n\) |

The final row has a polynomial of degree \(m\) being divided by a polynomial of degree \(n\). Can you complete this row?

At the beginning we thought about numerical division and asked ‘Why will the remainder will always be less than the divisor?’ Are there any similarities between the remainder and divisor in algebraic division and the remainder and divisor in numerical division?

For G. we have written that the remainder is zero, rather than the degree of the polynomial. The zero polynomial has no nonzero terms so it can be said that the degree is undefined. However it is sometimes useful to define it to be \(-\infty\).

Without any calculations, what can we say about the degree of the quotient and remainder polynomials of the following division?

I. \(\dfrac{x^9 - 5x^7 + 3x^2 - 11x + 2}{2x^5 + 4x^4 - x + 7}\)

Is it possible to be any more precise about the degree of the remainder?

The quotient must have degree 4. We have been writing the answers to the above questions in the form \[\dfrac{p(x)}{d(x)} = q(x) + \dfrac{r(x)}{d(x)}\] where \(q(x)\) is the quotient and \(r(x)\) is the remainder. This could also be written as \[p(x) = q(x)d(x) + r(x).\]

Since the divisor has degree 5, then the quotient must have degree 4 in order to produce a polynomial of degree 9.