find the remainder when 2x^(3)-3x^(2)+4x-5 is divided by x-1

1 answer

To find the remainder when the polynomial \( 2x^3 - 3x^2 + 4x - 5 \) is divided by \( x - 1 \), we can use the Remainder Theorem. The theorem states that the remainder of a polynomial \( f(x) \) divided by \( x - c \) is simply \( f(c) \).

In this case, we set \( c = 1 \) because we are dividing by \( x - 1 \).

Let's evaluate the polynomial at \( x = 1 \):

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

Calculating each term:

\[ f(1) = 2(1) - 3(1) + 4 - 5 \] \[ = 2 - 3 + 4 - 5 \] \[ = (2 - 3) + (4 - 5) \] \[ = -1 - 1 \] \[ = -2 \]

Thus, the remainder when \( 2x^3 - 3x^2 + 4x - 5 \) is divided by \( x - 1 \) is \(\boxed{-2}\).