Sure, I can assist with that.
The problem asks us to perform the division of the polynomial (x³-3x²+4x-4) by (x-2). This process is similar to long division, except we're working with polynomial terms instead of plain numbers. The result of this division will be another polynomial, and possibly a remainder polynomial.
Here are the steps to perform the division:
1. Divide the first term of the numerator by the first term of the denominator: x³ ÷ x = x². This is the first term of our quotient.
2. Multiply back each term in the divisor by that first term from the quotient: x²(x - 2) = x³ - 2x².
3. Then, subtract this result from the original numerator to get a new polynomial: (x³-3x²+4x-4) - (x³ - 2x²) = -1x² + 4x - 4.
4. Next, we repeat this same process with this new polynomial: Divide the first term -1x² by the first term of the denominator x, giving us -x.
5. Multiply back each term in the divisor by this term from the quotient: -x(x - 2) = -x² + 2x.
6. Subtract this result from the new polynomial above: (-1x² + 4x - 4) - (-x² + 2x) = 2x - 4.
7. Again, we repeat these steps: Divide the first term of the new polynomial 2x by x in the denominator, giving us 2.
8. Multiply each term in the divisor by this term from the quotient: 2(x - 2) = 2x - 4.
9. Subtract this result from the new polynomial. This gives us (2x - 4) - (2x - 4) = 0.
That means there's no remainder, and the process stops here.
The final result, or the quotient of the division, is made up of the terms we found at each step: x² - x + 2.
So, the given polynomial (x³-3x²+4x-4) divided by (x-2) equals to (x² - x + 2) with 0 as a remainder.