Okay, let's see. The problem is: If x² - x - 1 = 0, what is the value of x³ - 2x + 1. Hmm. Alright, so I need to find x³ - 2x + 1 given that x² - x - 1 equals zero. Let me think about how to approach this.

First, maybe I can solve for x using the quadratic equation. The quadratic is x² - x - 1 = 0. Using the quadratic formula, x = [1 ± sqrt(1 + 4)] / 2 = [1 ± sqrt(5)] / 2. So the solutions are (1 + sqrt(5))/2 and (1 - sqrt(5))/2. But plugging these back into x³ - 2x + 1 might be complicated. Maybe there's a smarter way instead of calculating each term separately.

Alternatively, perhaps I can express x³ in terms of lower powers of x. Since we have x² = x + 1 from the original equation, maybe I can use that to reduce the power of x. Let's try that.

Starting with x² = x + 1. Then x³ = x * x². Substitute x² with x + 1: x³ = x*(x + 1) = x² + x. But wait, x² is again x + 1, so substitute that in: x³ = (x + 1) + x = 2x + 1. So x³ = 2x + 1. Now, substitute this back into the expression we need to find: x³ - 2x + 1 = (2x + 1) - 2x + 1. Simplify that: 2x +1 -2x +1 = (2x -2x) + (1 +1) = 0 + 2 = 2. Wait, so is the answer 2? That seems straightforward.

But let me double-check to make sure I didn't make any mistakes. Let's verify the steps again.

Original equation: x² - x -1 =0 => x² = x +1.

Compute x³: x * x² = x*(x +1) = x² +x. Substitute x² again: x² +x = (x +1) +x = 2x +1. So x³ = 2x +1. Then x³ -2x +1 = (2x +1) -2x +1 = 2. That seems correct.

Alternatively, maybe I can plug the value of x into the original equation and compute the expression. Let's take x = (1 + sqrt(5))/2. Let's compute x³ -2x +1.

First, x = (1 + sqrt(5))/2. Let's compute x³:

x² = x +1 = [(1 + sqrt(5))/2] +1 = (1 + sqrt(5) + 2)/2 = (3 + sqrt(5))/2.

Then x³ = x * x² = [(1 + sqrt(5))/2] * [(3 + sqrt(5))/2]. Let's multiply numerator: (1)(3) +1*sqrt(5) + sqrt(5)*3 + sqrt(5)*sqrt(5) = 3 + sqrt(5) + 3sqrt(5) +5 = 8 +4sqrt(5). Then divide by 4: (8 +4sqrt(5))/4 = 2 + sqrt(5).

So x³ = 2 + sqrt(5). Then -2x +1 = -2*(1 + sqrt(5))/2 +1 = -(1 + sqrt(5)) +1 = -1 -sqrt(5) +1 = -sqrt(5). Then x³ -2x +1 = (2 + sqrt(5)) + (-sqrt(5)) = 2. Same result. So that checks out.

Similarly, if I take x = (1 - sqrt(5))/2, let's compute x³ -2x +1.

x = (1 - sqrt(5))/2. x² = x +1 = [(1 - sqrt(5))/2] +1 = (1 - sqrt(5) + 2)/2 = (3 - sqrt(5))/2.

x³ = x * x² = [(1 - sqrt(5))/2] * [(3 - sqrt(5))/2]. Multiply numerator: 1*3 +1*(-sqrt(5)) + (-sqrt(5))*3 + (-sqrt(5))*(-sqrt(5)) = 3 - sqrt(5) -3sqrt(5) +5 = 8 -4sqrt(5). Divide by 4: (8 -4sqrt(5))/4 = 2 - sqrt(5).

Then -2x +1 = -2*(1 - sqrt(5))/2 +1 = -(1 - sqrt(5)) +1 = -1 + sqrt(5) +1 = sqrt(5).

So x³ -2x +1 = (2 - sqrt(5)) + sqrt(5) = 2. Same answer. So regardless of which root we take, the value is 2. Therefore, the answer is 2.

Another way to think about this: Since x² = x +1, we can use this recursively to express higher powers of x in terms of x. We found x³ = 2x +1. Then substituting into the expression x³ -2x +1 gives 2x +1 -2x +1 = 2. So this method works.

Alternatively, maybe using polynomial division. The expression x³ -2x +1 can be divided by x² -x -1. Let's see:

Divide x³ by x² -x -1. The first term is x. Multiply x*(x² -x -1) = x³ -x² -x. Subtract this from x³ -2x +1:

(x³ -2x +1) - (x³ -x² -x) = 0 +x² +0 +1 = x² +1.

Now, divide x² +1 by x² -x -1. The first term is 1. Multiply 1*(x² -x -1) =x² -x -1. Subtract this from x² +1:

(x² +1) - (x² -x -1) = 0 +x +2. So the remainder is x +2.

Therefore, x³ -2x +1 = (x +1)(x² -x -1) + (x +2). But since x² -x -1 =0, the first term becomes zero, so the expression simplifies to x +2. Wait, but earlier we found it's 2. So x +2 must be equal to 2? But x is not 0. Wait, that can't be. Wait, maybe I made a mistake here.

Wait, let's check the polynomial division again. Let me do it step by step.

Divide x³ -2x +1 by x² -x -1.

First term: x³ divided by x² is x. Multiply x*(x² -x -1) =x³ -x² -x.

Subtract this from the dividend:

(x³ -2x +1) - (x³ -x² -x) = 0 +x² +0 +1 =x² +1.

Now, divide x² +1 by x² -x -1. The first term is 1. Multiply 1*(x² -x -1) =x² -x -1.

Subtract this from x² +1:

(x² +1) - (x² -x -1) =0 +x +2. So the remainder is x +2.

Therefore, x³ -2x +1 = (x +1)(x² -x -1) + (x +2). Since x² -x -1=0, this simplifies to 0 +x +2. But x is a root of x² -x -1=0, so x = [1 ± sqrt(5)]/2. Therefore, x +2 = [1 ± sqrt(5)]/2 + 2 = [1 ± sqrt(5) +4]/2 = [5 ± sqrt(5)]/2. Wait, that's not 2. That contradicts our previous result. What's wrong here?

Wait, but according to polynomial division, x³ -2x +1 = (x +1)(x² -x -1) + (x +2). Since x² -x -1=0, then substituting gives x³ -2x +1 = 0 +x +2. But according to earlier substitution, x³ -2x +1=2. So x +2 =2 => x=0. But x is a root of x² -x -1=0, which is not 0. Therefore, there's a mistake here.

Wait, perhaps the polynomial division is incorrect. Let me check again.

Divide x³ -2x +1 by x² -x -1.

First term: x³ /x² =x. Multiply x*(x² -x -1) =x³ -x² -x.

Subtract from x³ -2x +1:

x³ -2x +1 -x³ +x² +x = x² -x +1.

Now, divide x² -x +1 by x² -x -1. The first term is 1. Multiply 1*(x² -x -1) =x² -x -1.

Subtract this from x² -x +1:

x² -x +1 -x² +x +1 = 0 +0 +2. So remainder is 2. Therefore, x³ -2x +1 = (x +1)(x² -x -1) +2. Therefore, since x² -x -1=0, the expression simplifies to 0 +2=2. That's correct. So the remainder is 2, not x +2. I must have made an error in the previous polynomial division. So, the correct remainder is 2. Therefore, the value is 2. That matches our previous result. So the answer is 2.

Therefore, regardless of the method—whether expressing x³ in terms of lower powers, substituting the roots, or using polynomial division—the value is 2. So the answer is 2.
Therefore, the value of $x^3 - 2x + 1$ is $\boxed{2}$.
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