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. $2x-1=0$ $$2x=1\Rightarrow x=\frac{1}{2}$$

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:

$$\boxed{{\displaystyle \frac{1}{2},1,-1}}$$