Simplify:
(x3 − 2x2 + 3x − 4) (x −1) − (2x − 3)(x2 − x + 1)
Solution
To simplify,we will proceed as follows:
\[\left( x^3 - 2 x^2 + 3x - 4 \right)\left( x - 1 \right) - \left( 2x - 3 \right)\left( x^2 - x + 1 \right)\]
\[ = \left[ \left( x^3 - 2 x^2 + 3x - 4 \right)\left( x - 1 \right) \right] - \left[ \left( 2x - 3 \right)\left( x^2 - x + 1 \right) \right]\]
\[= \left[ x\left( x^3 - 2 x^2 + 3x - 4 \right) - 1\left( x^3 - 2 x^2 + 3x - 4 \right) \right] - \left[ 2x\left( x^2 - x + 1 \right) - 3\left( x^2 - x + 1 \right) \right]\] (Distributive law)
\[= \left[ x\left( x^3 - 2 x^2 + 3x - 4 \right) - 1\left( x^3 - 2 x^2 + 3x - 4 \right) \right] - \left[ 2x\left( x^2 - x + 1 \right) - 3\left( x^2 - x + 1 \right) \right]\]
\[ = x^4 - 2 x^3 + 3 x^2 - 4x - x^3 + 2 x^2 - 3x + 4 - \left[ 2 x^3 - 2 x^2 + 2x - 3 x^2 + 3x - 3 \right]\]
\[ = x^4 - 2 x^3 + 3 x^2 - 4x - x^3 + 2 x^2 - 3x + 4 - 2 x^3 + 2 x^2 - 2x + 3 x^2 - 3x + 3\]
\[= x^4 - 2 x^3 - 2 x^3 - x^3 + 3 x^2 + 2 x^2 + 2 x^2 + 3 x^2 - 4x - 3x - 2x - 3x + 4 + 3\]
(Rearranging)
\[= x^4 - 5 x^3 + 10 x^2 - 12x + 7\] (Combining like terms)
Thus, the answer is \[x^4 - 5 x^3 + 10 x^2 - 12x + 7\].
