What is the remainder when \( x^3 - 2x^2 + 3x - 4 \) is divided by \( x - 1 \)?

To find the remainder when ( x^3 - 2x^2 + 3x - 4 ) is divided by ( x - 1 ), you can apply the Remainder Theorem. This theorem states that the remainder of the division of a polynomial ( f(x) ) by ( x - k ) is simply ( f(k) ).

Here, ( f(x) = x^3 - 2x^2 + 3x - 4 ) and we are dividing by ( x - 1 ), which means we should evaluate ( f(1) ).

Calculating ( f(1) ):

1. Substitute ( 1 ) into the polynomial:

[

f(1) = (1)^3 - 2(1)^2 + 3(1) - 4

]

2. Simplify each term:

[

= 1 - 2 + 3 - 4

]

3. Combine the values:

[

= 1 - 2 = -1

]

[

-1 + 3 = 2

]

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