We are asked to compare and contrast dividing real numbers and dividing polynomials. We will focus on the long division method for each of these cases. Let's recall the steps taken in the long division of real numbers and polynomial long division using appropriate examples. Then we can form our conclusion.

Long Division of Real Numbers

Let's divide $243$ by $7$ using long division. Starting from the left-hand side of the $dividend,$ we want to identify the shortest sequence of digits that form a number greater than the $divisor .$

In this case, the number is $24 .$ Next, we should identify the largest whole number that is the multiple of the $divisor$ and is less than $24 .$ The number meeting these conditions is $21 = 3 \times 7 .$ The digit $3$ is put above the $dividend .$ It is the first digit of the quotient.

This step can be also thought of as dividing $24$ by $7$ and using the whole part of the result as the first digit of the quotient. Now let's multiply the $first$ $digit$ $of$ $the$ $quotient$ by the $divisor$ and write the product under the $dividend .$

We will subtract the product from the $dividend .$ When doing so, we will only consider the digits of the $dividend$ that formed a number greater than the $divisor .$

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