Use factoring by grouping to solve the equation.

2 x cubed minus x squared minus 2 x plus 1 equals 0
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Part 1
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To solve the equation 2x3x22x+1=0 using factoring by grouping, we start by grouping the terms in pairs:

(2x3x2)+(2x+1)

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

  1. From the first group 2x3x2, we can factor out x2: x2(2x1)

  2. From the second group 2x+1, we can factor out 1: 1(2x1)

Now we can combine these factored groups:

x2(2x1)1(2x1)=0 (2x1)(x21)=0

Next, we can further factor x21 using the difference of squares:

x21=(x1)(x+1)

Thus, we have:

(2x1)(x1)(x+1)=0

Setting each factor equal to zero gives:

  1. 2x1=0 2x=1x=12

  2. x1=0 x=1

  3. x+1=0 x=1

Thus, the solutions to the equation 2x3x22x+1=0 are:

12,1,1

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