(4x^3-3x^2+7x)/(x+1/2)

Quotient and remainder

hard to show long algebraic division, but I got

= 4x^2 - 5x + 8.5 with a remainder of 4.75

same result using synthetic division

-.5 | 4 -3 7 0

......... -2 2.5 4.75
...... 4 -5 9.5 4.75

A good site that includes polynomial long division, showing the full details is

calc101 . com

To find the quotient and remainder of the given polynomial division, we can use polynomial long division method. Here's how:

Step 1: Set up the division problem
We have: (4x^3 - 3x^2 + 7x) ÷ (x + 1/2)

Step 2: Arrange the terms in descending order of degree
Rewrite the dividend in descending order of powers of x:
4x^3 - 3x^2 + 7x + 0

Step 3: Divide the highest-degree term of the dividend by the highest-degree term of the divisor
Divide 4x^3 by x to get 4x^2.

Step 4: Multiply the divisor by the quotient obtained in the previous step
Multiply (x + 1/2) by 4x^2 to get 4x^3 + 2x^2.

Step 5: Subtract the product obtained in the previous step from the dividend
Subtract (4x^3 + 2x^2) from (4x^3 - 3x^2 + 7x) to get -5x^2 + 7x.

Step 6: Bring down the next term from the dividend
Bring down the next term, which is 7x.

Step 7: Repeat steps 3 to 6 until you cannot divide anymore
Divide (-5x^2 + 7x) by (x + 1/2) to get -5x + 6.

Step 8: Write the quotient and remainder
The quotient is 4x^2 - 5x + 6, and the remainder is 0.

Therefore, the quotient of (4x^3 - 3x^2 + 7x) divided by (x + 1/2) is 4x^2 - 5x + 6, and the remainder is 0.