Factor completely. \( 17 y^{3}-34 y^{2}+12 y-24 \)

To factor the polynomial \( 17 y^{3}-34 y^{2}+12 y-24 \), we can start by grouping the terms. First, we notice that we can factor out a common factor of \( 17 \) from the first two terms and \( 12 \) from the last two terms: \[ 17(y^3 - 2y^2) + 12(y - 2) \] However, this doesn't seem to help with the grouping approach immediately, but let's factor it another way. Let's look for rational roots using the Rational Root Theorem. After some inspection, we find that \( y = 2 \) is a root. Using synthetic division: 1. Set up the synthetic division with \( y - 2 \). 2. Divide \( 17 y^3 - 34 y^2 + 12 y - 24 \) by \( (y - 2) \). We get: \[ 17 y^{2} + 0y + 12 \] So, the polynomial can be expressed as: \[ (y - 2)(17y^2 + 12) \] Next, the quadratic \( 17y^2 + 12 \) doesn’t factor nicely over the integers. Thus, the completely factored form of the polynomial is: \[ (y - 2)(17y^2 + 12) \] And that’s your answer!