If divided by (x - 3) the polynomials x ^ 3 - k * x ^ 2 + 6 and 2x ^ 3 - x ^ 2 + (k + 3) - 6 leave the same remainder. Find the value of k.

To find the value of \( k \) such that the polynomials \( P(x) = x^3 - kx^2 + 6 \) and \( Q(x) = 2x^3 - x^2 + (k + 3)x - 6 \) leave the same remainder when divided by \( x - 3 \), we need to find the remainders of these polynomials when \( x = 3 \).

The Remainder Theorem states that the remainder of the division of a polynomial \( P(x) \) by \( x - a \) is \( P(a) \).

First, we find the remainder of \( P(x) = x^3 - kx^2 + 6 \) when \( x = 3 \):

\[ P(3) = 3^3 - k \cdot 3^2 + 6 = 27 - 9k + 6 = 33 - 9k \]

Next, we find the remainder of \( Q(x) = 2

Discover how to find the value of k in polynomials using the Remainder Theorem. Solve the problem where two polynomials leave the same remainder when divided by (x - 3). This problem involves advanced polynomial concepts and is suitable for students in Grades 10-12.

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