We are asked to compare and contrast dividing and dividing . 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 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 that form a number greater than the divisor.
In this case, the number is 24. Next, we should identify the largest that is the multiple of the divisor and is less than 24. The number meeting these conditions is 21= 3*7. The digit 3 is put above the dividend. It is the first digit of the .
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 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.
Finally, we will bring down the next digit of the dividend. The result of this operation is called the remainder.
To find the second digit of the quotient, we will follow the same steps treating the remainder as the dividend.
In this case, there are no more digits of the dividend that we could bring down. When the remainder is less than the divisor we have two options.
- Stop dividing.
- Put a decimal point at the end of our current quotient, add a 0 to the remainder, and continue dividing.
Using our example, we can list the steps for the long division of real numbers.
- Identify the shortest sequence of digits of the dividend that form a number greater than the divisor.
- Divide the number from Step 1 by the divisor. The whole part of the result is the first digit of the quotient.
- Multiply the first digit of the quotient by the divisor and write the product under the dividend.
- Subtract the product from the dividend.
- Bring down the next digit of the dividend. The result of this operation is a remainder.
- Repeat Steps 2 through 5 treating the remainder as the dividend.
Long Division of Polynomials
Assume we want to divide 2x^3-7x^2+1 by x+1 using polynomial long division. Before the actual division can be performed, we have to make sure that the dividend and divisor are written in . If a is not present in the dividend, we should add the term with a of 0.
Dividend: & 2x^3-7x^2+1
Divisor: & x+1
The terms of each polynomial are in descending order, so they are already written in standard form. However, the dividend has one missing term.
2x^3-7x^2+1 ⇔ 2x^3-7x^2+ 0x+1
Now, we are ready to divide. Let's find the first term of the quotient. To do so, we will divide the first term of the dividend by the first term of the divisor.
2x^3/x=2x^2
This is the first term of the quotient which is noted as shown below.
Next we will multiply the first term of the quotient by the divisor.
2x^2(x+1)
2x^2* x+2x^2* 1
2x^3+2x^2* 1
2x^3+2x^2
Finally, we will bring down the next terms of the dividend. The result of this operation is called the remainder.
To find the second term of the quotient, we will treat the remainder as the dividend and follow the same steps as before.
The third term of the quotient can be found by following the same steps once more. This time the remainder 9x+1 will be treated as the dividend.
The remainder - 8 is a , so its degree is 0. The degree of the divisor x+1 is 1. Since the degree of the remainder is less than the degree of the divisor, this is the end of the division. Using our example, we can list the steps of polynomial long division.
- Write the dividend and divisor in standard form. If a term is not present in the dividend, add the term with a coefficient of 0.
- Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient.
- Multiply the first term of the quotient by the divisor and write the product under the dividend.
- Subtract the product from the dividend.
- Bring down the next terms of the dividend. The result of this operation is a remainder.
- Repeat Steps 2 through 5 treating the remainder as the dividend. Stop when the degree of the remainder is less than the degree of the divisor.
Conclusion
Long division of real numbers and long division of polynomials can be both described in six steps. Both methods share the steps of divide, multiply, subtract, and bring down. The difference between them is that the digits are replaced by the terms of the polynomials. Additionally, how we start and when we stop the division differs.
