Use factoring by grouping to solve the equation.

Question

2 x cubed minus x squared minus 2 x plus 1 equals 0
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Part 1
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Answer 1

To solve the equation \(2x^3 - x^2 - 2x + 1 = 0\) using factoring by grouping, we start by grouping the terms in pairs:

\[ (2x^3 - x^2) + (-2x + 1) \]

Next, we factor out the common factors from each group:

1. From the first group \(2x^3 - x^2\), we can factor out \(x^2\): \[ x^2(2x - 1) \]

2. From the second group \(-2x + 1\), we can factor out \(-1\): \[ -1(2x - 1) \]

Now we can combine these factored groups:

\[ x^2(2x - 1) - 1(2x - 1) = 0 \] \[ (2x - 1)(x^2 - 1) = 0 \]

Next, we can further factor \(x^2 - 1\) using the difference of squares:

\[ x^2 - 1 = (x - 1)(x + 1) \]

Thus, we have:

\[ (2x - 1)(x - 1)(x + 1) = 0 \]

Setting each factor equal to zero gives:

1. \(2x - 1 = 0\) \[ 2x = 1 \quad \Rightarrow \quad x = \frac{1}{2} \]

2. \(x - 1 = 0\) \[ x = 1 \]

3. \(x + 1 = 0\) \[ x = -1 \]

Thus, the solutions to the equation \(2x^3 - x^2 - 2x + 1 = 0\) are:

\[ \boxed{\frac{1}{2}, 1, -1} \]