Answer: \(2y^3 + y^2 - 2y - 1\) factors to \((2y + 1)(y - 1)(y + 1)\).
Step-by-step explanation:To factorize the polynomial \(2y^3 + y^2 - 2y - 1\), we'll use a technique called grouping.
1. Grouping Technique:
We'll group pairs of terms together and factor out common factors.
\(2y^3 + y^2 - 2y - 1\)
Group the first two terms and the last two terms:
\((2y^3 + y^2) + (-2y - 1)\)
2. Factor Common Terms from Each Group:
From the first group, factor out \(y^2\), and from the second group, factor out \(-1\):
\(y^2(2y + 1) - 1(2y + 1)\)
3. Observe a Common Factor:
Now, we see that \((2y + 1)\) is a common factor:
\((2y + 1)(y^2 - 1)\)
4. Factor the Difference of Squares:
\(y^2 - 1\) is a difference of squares, which can be factored further:
\((2y + 1)(y - 1)(y + 1)\)
So, \(2y^3 + y^2 - 2y - 1\) factors to \((2y + 1)(y - 1)(y + 1)\).