X2-x3+2x2+x+1 by X-1 Synthetic method
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
We want to divide the polynomial $x^3 - x^2 + 2x^2 + x + 1$ by $x-1$. First, simplify the polynomial: $x^3 + x^2 + x + 1$.
Using synthetic division with the root $1$ from the divisor $x-1$:
`
1 | 1 1 1 1
| 1 2 3
----------------
1 2 3 4
`
The coefficients of the quotient are $1, 2, 3$, and the remainder is $4$.
The quotient is $x^2 + 2x + 3$ and the remainder is $4$.
So, $\frac{x^3 + x^2 + x + 1}{x-1} = x^2 + 2x + 3 + \frac{4}{x-1}$.
The final answer is $\boxed{x^2 + 2x + 3 + \frac{4}{x-1}}$.