What should be added to `x^(3)-x^(2) -x+1` to get `5x^(3) -4x^(2) +3x -1`?

Answer

To find what should be added to the polynomial \( x^3 - x^2 - x + 1 \) to obtain \( 5x^3 - 4x^2 + 3x - 1 \), we can set up the equation:

\[ x^3 - x^2 - x + 1 + A = 5x^3 - 4x^2 + 3x - 1 \]

Where \( A \) is the polynomial we need to find. Rearranging the equation gives:

\[ A = (5x^3 - 4x^2 + 3x - 1) - (x^3 - x^2 - x + 1) \]

Now, we will simplify the right-hand side:

1. Distribute the negative sign across the second polynomial:

\[ A = 5x^3 - 4x^2 + 3x - 1 - x^3 + x^2 + x - 1 \]

2. Combine like terms:

- For \( x^3 \): \( 5x^3 - x^3 = 4x^3 \)
- For \( x^2 \): \( -4x^2 + x^2 = -3x^2 \)
- For \( x \): \( 3x + x = 4x \)
- For the constant term: \( -1 - 1 = -2 \)

Putting it all together, we get:

\[ A = 4x^3 - 3x^2 + 4x - 2 \]

Thus, the polynomial that should be added is:

\[ \boxed{4x^3 - 3x^2 + 4x - 2} \]