Bot GPT 4-o mini To find the remainder when the polynomial \( x^3 + 3x^2 - 10x - 14 \) is divided by \( x - 3 \), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the polynomial \( f(x) \) when divided by \( x - c \) is \( f(c) \).
In this case, our polynomial is
\[ f(x) = x^3 + 3x^2 - 10x - 14 \]
We need to find \( f(3) \):
\[ f(3) = 3^3 + 3(3^2) - 10(3) - 14 \]
Calculating each term:
\[ 3^3 = 27 \] \[ 3(3^2) = 3(9) = 27 \] \[ -10(3) = -30 \]
Now substituting these values into \( f(3) \):
\[ f(3) = 27 + 27 - 30 - 14 \]
Now, perform the additions and subtractions:
\[ 27 + 27 = 54 \] \[ 54 - 30 = 24 \] \[ 24 - 14 = 10 \]
Thus, the remainder when \( x^3 + 3x^2 - 10x - 14 \) is divided by \( x - 3 \) is
\[ \boxed{10} \]
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