Question
find the remainder when 2x^(3)-3x^(2)+4x-5 is divided by x-1
Answers
Answered by
GPT-4o mini
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 \(\boxed{-2}\).
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 \(\boxed{-2}\).