# Complete the Division Problem. What Is the Remainder? -18X – 7 2X + 3 -2X + 5 6X + 5

Complete the Division Problem: What Is the Remainder? -18X – 7 2X + 3 -2X + 5 6X + 5

Division is a fundamental concept in mathematics that involves breaking down a number or expression into smaller, equal parts. When we divide one expression by another, we often encounter remainders. In this article, we will explore the concept of division, specifically focusing on the division problem -18X – 7 ÷ 2X + 3 -2X + 5 ÷ 6X + 5, and understand what the remainder represents.

To begin with, let’s break down the division problem step by step:

Step 1: -18X – 7 ÷ 2X + 3
Step 2: -2X + 5 ÷ 6X + 5

Step 1: -18X – 7 ÷ 2X + 3

In this step, we are dividing -18X – 7 by 2X + 3. To simplify the division, we can use long division or synthetic division. Let’s use long division for this example:

-9X – 4
__________
2X + 3 | -18X – 7

-18X divided by 2X gives us -9, which we write above the division symbol. Multiplying 2X + 3 by -9X gives us -18X^2 – 27X. Subtracting this from -18X – 7 gives us 20X – 7.

-9X – 4
__________
2X + 3 | -18X – 7
-18X – 27
_______________
20X – 7

Now, we bring down the next term, which is -2X. We divide 20X by 2X, giving us 10. Multiplying 2X + 3 by 10 gives us 20X + 30. Subtracting this from 20X – 7 gives us -37.

-9X – 4
__________
2X + 3 | -18X – 7
-18X – 27
_______________
20X – 7
20X + 30
_______________
-37

We have reached the end of the division problem for the first step. The remainder in this case is -37.

Step 2: -2X + 5 ÷ 6X + 5

Similarly, let’s divide -2X + 5 by 6X + 5. Again, we can use long division:

-1/3
_________
6X + 5 | -2X + 5

-2X divided by 6X gives us -1/3, which we write above the division symbol. Multiplying 6X + 5 by -1/3 gives us -2X – 5/3. Subtracting this from -2X + 5 gives us 20/3.

-1/3
_________
6X + 5 | -2X + 5
-2X – 5/3
_______________
20/3

In the second step, the remainder is 20/3.

FAQs

Q1: What is a remainder in division?
A1: A remainder is the amount left over when one number or expression is divided by another number or expression. It represents the part that cannot be evenly divided.

Q2: Can a remainder be negative?
A2: Yes, a remainder can be negative. It depends on the numbers or expressions being divided and the division process.

Q3: How can we determine the remainder in division?
A3: To determine the remainder, we divide one number or expression by another and subtract the product of the divisor and the quotient from the dividend. The result is the remainder.

Q4: Can a remainder be a fraction or decimal?
A4: Yes, a remainder can be a fraction or decimal if the division does not yield a whole number quotient. The remainder can be expressed as a fraction or decimal.

In conclusion, division is a mathematical operation that involves breaking down numbers or expressions into smaller, equal parts. The remainder in a division problem represents the leftover part that cannot be evenly divided. By solving the division problem -18X – 7 ÷ 2X + 3 -2X + 5 ÷ 6X + 5, we can determine the remainder at each step. The remainder can be positive, negative, a fraction, or a decimal, depending on the specific division problem.