Please correct my work. Using long division,simplify (8b^3-6),devided by (2b-1)
My solution is:
4b^2+2b+1
----------------
8b^3 -6
8b^3-4b^2
-----------------
4b^2-6
4b^2-2b
------------------
2b-6
2b-1
-------------------
-5
( 8 b ^ 3 - 6 ) / ( 2 b - 1 ) =
4 b ^ 2 + 2 b + 1 - 5 / ( 2 b - 1 )
a good place to check your work is
http://calc101.com/webMathematica/long-divide.jsp
To simplify the expression (8b^3 - 6) divided by (2b - 1) using long division, follow these steps:
Step 1: Start by dividing the highest degree term of the numerator (8b^3) by the highest degree term of the denominator (2b). This will give you the first term of the quotient, which is 4b^2.
Step 2: Multiply the entire denominator (2b - 1) by the term you found in step 1 (4b^2). You will get 8b^3 - 4b^2.
Step 3: Subtract the product from step 2 (8b^3 - 4b^2) from the original numerator (8b^3 - 6). This will give you a new polynomial, which you will then use to repeat the process.
Subtraction: (8b^3 - 4b^2) - (8b^3 - 6) = -4b^2 + 6.
Step 4: Bring down the next term from the original numerator, which is -4b^2 + 6.
Step 5: Divide the highest degree term of the new polynomial (-4b^2) by the highest degree term of the denominator (2b). This gives you -2b.
Step 6: Multiply the entire denominator (2b - 1) by the term you found in step 5 (-2b). You will get -4b^2 + 2b.
Step 7: Subtract the product from step 6 (-4b^2 + 2b) from the new polynomial (-4b^2 + 6). This will give you a new polynomial, which you will then use to repeat the process.
Subtraction: (-4b^2 + 2b) - (-4b^2 + 6) = -4b^2 + 2b - (-4b^2) + 6 = 4b - 4.
Step 8: Bring down the next term from the original numerator, which is 4b - 4.
Step 9: Divide the highest degree term of the new polynomial (4b) by the highest degree term of the denominator (2b). This gives you 2.
Step 10: Multiply the entire denominator (2b - 1) by the term you found in step 9 (2). You will get 4b - 2.
Step 11: Subtract the product from step 10 (4b - 2) from the new polynomial (4b - 4). This will give you a new polynomial, which is the remainder.
Subtraction: (4b - 4) - (4b - 2) = -2.
Step 12: The remainder is -2.
Therefore, the simplified expression is 4b^2 - 2b + 2 - (2 / (2b - 1)).