There is an error in the student work shown below:
Question: Divide (x^3-8)/(x-2)
The student did long division and the got x^2 remainder -6+2x^2
Yup, there is an error!
However, it is a poor question which asks to explain somebody's error
Correct way:
(x^3-8) = (x-2)(x^2 + 2x+ 4)
so ...
(x^3-8)(x-2)
= (x-2)(x^2 + 2x+ 4)/(x-2)
= x^2 + 2x + 4 , x ≠ 2
To find the error in the student's work, we need to perform the long division correctly and compare the results.
Step 1: Divide x^3 by x to get x^2.
(x^2 - 8)/(x - 2)
Step 2: Multiply (x - 2) by x^2 to get x^3 - 2x^2.
Subtract this result from the original polynomial to get:
(x^3 - 8) - (x^3 - 2x^2) = -2x^2 - 8
Step 3: Bring down the next term (-2x^2).
(-2x^2 - 8)/(x - 2)
Step 4: Divide -2x^2 by x to get -2x.
Multiply (x - 2) by -2x to get -2x^2 + 4x.
Subtract this result from the previous step:
(-2x^2 - 8) - (-2x^2 + 4x) = -12 - 4x.
Step 5: Bring down the next term (-12).
(-12 - 4x)/(x - 2)
Step 6: Divide -12 by x to get -12/x.
Multiply (x - 2) by -12/x to get -12 + 24/x.
Subtract this result from the previous step:
(-12 - 4x) - (-12 + 24/x) = -4x + 24/x.
The correct result of the long division is:
(x^2 - 2x - 4) with a remainder of (-4x + 24/x).
The student's error was in Step 2, where they incorrectly subtracted (x^3 - 2x^2) from (x^3 - 8). The correct subtraction should result in -2x^2 - 8, not -6 + 2x^2.
To check for errors in the student's work, we can perform the long division ourselves and compare the results.
To divide (x^3 - 8) by (x - 2), we follow these steps:
1. Set up the division problem:
________________________
(x - 2) | (x^3 - 8)
2. Divide the first term of the numerator (x^3) by the first term of the denominator (x):
(x^3 / x) = x^2
3. Multiply the entire denominator (x - 2) by the quotient we obtained (x^2) and write the result below the numerator.
(x^2) * (x - 2) = x^3 - 2x^2
4. Subtract the result obtained in step 3 from the numerator (x^3 - 8) and write the difference below the line.
(x^3 - 8) - (x^3 - 2x^2) = -8 + 2x^2
At this point, we have a remainder (-8 + 2x^2), which means the division is not complete. The student's answer of "x^2 remainder -6 + 2x^2" seems to be incorrect.
To verify this, we can continue the division:
5. Bring down the next term from the numerator (-8 + 2x^2), which is -6:
-6
6. Divide the new numerator (-6) by the denominator (x - 2):
(-6 / (x - 2)) = -6/(x - 2)
So, the correct result of the division is x^2 - 6/(x - 2), without any remainder.
Therefore, the student made a mistake in their long division, resulting in an incorrect remainder.