Find the remainder when \[ P(x)=2 x^{3}+2 x^{2}-10 x+6 \] is divided by \( (x-2) \) without performing long division.

To find the remainder when \( P(x) = 2x^3 + 2x^2 - 10x + 6 \) is divided by \( (x-2) \), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( P(x) \) by \( (x-c) \) is simply \( P(c) \). In our case, we are dividing by \( (x-2) \), so we need to evaluate \( P(2) \). Calculating \( P(2) \): \[ P(2) = 2(2)^3 + 2(2)^2 - 10(2) + 6 \] Calculating each term: - \( 2(2)^3 = 2 \times 8 = 16 \) - \( 2(2)^2 = 2 \times 4 = 8 \) - \( -10(2) = -20 \) - The constant term is \( 6 \) Now, substitute these values back into the equation: \[ P(2) = 16 + 8 - 20 + 6 \] Combining these: \[ P(2) = 16 + 8 = 24 \] \[ 24 - 20 = 4 \] \[ 4 + 6 = 10 \] Thus, the remainder when \( P(x) \) is divided by \( (x-2) \) is \[ \boxed{10} \]