### Polynomials & Rational Functions

Package of problems

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## Problem

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

where $4$ is the divisor, $6$ is the quotient, and $3$ is the remainder.

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

Why will the remainder 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 $d(x)$?

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

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

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

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