(4x^3-4x^2-x-3)divided by(2x^2-3)
calc101 has many good tools, among which is polynomial division.
It will show that the answer is
2x-2
oh yeah - remainder 5x-9
To divide (4x^3 - 4x^2 - x - 3) by (2x^2 - 3), we can use polynomial long division. Here's how you can do it step by step:
Step 1: First, write the dividend (4x^3 - 4x^2 - x - 3) and the divisor (2x^2 - 3) in descending order of exponents:
4x^3 - 4x^2 - x - 3 ÷ 2x^2 - 3
Step 2: Divide the leading term of the dividend (4x^3) by the leading term of the divisor (2x^2). The result becomes the first term of the quotient.
The quotient so far: 2x
Step 3: Multiply the divisor (2x^2 - 3) by the first term of the quotient (2x). Place the product (4x^3 - 6x) under the dividend, aligning the like terms.
Subtract the product from the dividend:
4x^3 - 4x^2 - x - 3
- (4x^3 - 6x)
-------------
2x^2 - x - 3
Step 4: Bring down the next term from the dividend (in this case, -x). We now have the new dividend to work with.
The current dividend: 2x^2 - x - 3
Step 5: Divide the new leading term of the current dividend (2x^2) by the leading term of the divisor (2x^2). The result becomes the second term of the quotient.
The quotient so far: 2x - 1
Step 6: Multiply the divisor (2x^2 - 3) by the second term of the quotient (-1). Place the product (-2x^2 + 3) under the current dividend, aligning the like terms.
Subtract the product from the current dividend:
2x^2 - x - 3
- (2x^2 - 3)
-------------
2x - 6
Step 7: Bring down the next term from the current dividend (in this case, -6). We now have the new current dividend to work with.
The current dividend: 2x - 6
Step 8: Divide the new leading term of the current dividend (2x) by the leading term of the divisor (2x^2). The result becomes the third term of the quotient.
The quotient so far: 2x - 1 + 1/(2x - 3)
Step 9: Multiply the divisor (2x^2 - 3) by the third term of the quotient (1). Place the product (2x^2 - 3) under the current dividend, aligning the like terms.
Subtract the product from the current dividend:
2x - 6
- (2x - 3)
------------
-3
Step 10: There are no more terms left to bring down. The remainder is -3.
Therefore, the division of (4x^3 - 4x^2 - x - 3) by (2x^2 - 3) is equal to the quotient 2x - 1 + 1/(2x - 3) with a remainder of -3, represented as:
(4x^3 - 4x^2 - x - 3) ÷ (2x^2 - 3) = 2x - 1 + 1/(2x - 3) - 3/(2x^2 - 3)