Introduction

When dealing with integers, we say that $7$ is a factor of $56$ because

56 = 7 \times 8

and so $56$ can be divided by $7$ with no remainder.

The relationship with $10$ and $56$ is different. The closest we can get without exceeding $56$ is $10 \times 5 = 50,$ and we have a remainder of $6,$

56 = 10 \times 5 + 6

In this case, $6$ is the remainder and $5$ is called the quotient.

With polynomials, it is similar: $(x - 4)$ is not a factor of $x^{3} + 2 x^{2} - 11 x - 12,$ but we can write

x^{3} + 2 x^{2} - 11 x - 12 = (x - 4) (x^{2} + 6 x + 13) + 40

In this case, the $40$ is the remainder and $x^{2} + 6 x + 13$ is the quotient.

Over the page, we will learn how to find the quotient and remainder using long division.

The polynomial
$x^{4} + 2 x^{3} - 3 x - 1$
can be written in the form
$(x + 3) q (x) + r$
where $q (x)$ is a polynomial and $r$ is a constant.
Find $q (x)$ and $r .$
Use the factor theorem to show that $(x + 3)$ is a factor of $x^{3} + 4 x^{2} + x - 6$
Use the remainder theorem to find the remainder when $x^{4} - 2 x^{2} + x + 1$ is divided by $(x + 1)$
The remainder when
$p (x) = 2 x^{3} + a x^{2} - b x - 4$
is divided by $(x + 3)$ is equal to $- 25$ . Given also that $(x - 2)$ is a factor of $p,$ find the values of $a$ and $b .$

Introduction ​

Introduction