Divisibility of x³-x²-ax+b by x²-x

To find the values of a and b, we need to use the concept of polynomial division. We divide x³-x²-ax+b by x²-x, and if the result is a polynomial without any remainder, then x²-x is a factor of x³-x²-ax+b.

Polynomial Division

The steps for polynomial division are as follows:
1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
2. Multiply the result obtained in step 1 by the divisor and subtract it from the dividend.
3. Bring down the next term of the dividend and repeat steps 1 and 2 until all terms have been considered.

Calculating a and b

Let's apply polynomial division to x³-x²-ax+b divided by x²-x:
```
x
___________
x²-x | x³-x²-ax+b
- x³-x²
-------
ax+b
- ax+a
-------
b-a
```
The result of the division is ax+a+b/(x²-x), and there is no remainder. Therefore, x²-x is a factor of x³-x²-ax+b, and a and b can be calculated as follows:
- The coefficient of x in the quotient is a+1, so a = -1.
- The constant term in the quotient is b-a, so b = a-1 = -2.

Therefore, a = -1 and b = -2.

Conclusion

The values of a and b for x³-x²-ax+b to be divisible by x²-x are a = -1 and b = -2. Polynomial division is a useful tool for finding factors of polynomials and their corresponding coefficients.