Determining the Value of $x^3 + 1/x^3$

The problem asks us to find the value of $(x^3 + 1/x^3)$ given the equation $x^2 - 5x + 1 = 0$. This involves algebraic manipulation and the application of standard algebraic identities.

Finding the Value of $x + 1/x$

We start with the given equation: $x^2 - 5x + 1 = 0$ To find a relationship between $x$ and $1/x$, we can divide the entire equation by $x$. We can do this because $x=0$ is not a solution to the equation (substituting $x=0$ yields $1=0$, which is false). Dividing by $x$: $\frac{x^2}{x} - \frac{5x}{x} + \frac{1}{x} = \frac{0}{x}$ This simplifies to: $x - 5 + \frac{1}{x} = 0$ Rearranging the terms to isolate $x + 1/x$: $x + \frac{1}{x} = 5$ So, we have found that $x + 1/x = 5$.

Using the Sum of Cubes Identity

Our goal is to find $x^3 + 1/x^3$. We can use the algebraic identity for the sum of cubes:

$a^3 + b^3 = (a+b)^3 - 3ab(a+b)$

Let $a = x$ and $b = 1/x$. Substituting these into the identity gives:

$x^3 + \left(\frac{1}{x}\right)^3 = \left(x + \frac{1}{x}\right)^3 - 3(x)\left(\frac{1}{x}\right)\left(x + \frac{1}{x}\right)$

Simplify the identity: $x \times (1/x) = 1$, so the identity becomes:

$x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right)$

Calculating the Final Value

Now, we substitute the value we found earlier, $x + 1/x = 5$, into the equation:

$x^3 + \frac{1}{x^3} = (5)^3 - 3(5)$

Performing the calculations:

• Calculate $5^3$: $5^3 = 5 \times 5 \times 5 = 125$
• Calculate $3 \times 5$: $3 \times 5 = 15$

Substitute these values back:

$x^3 + \frac{1}{x^3} = 125 - 15$

$x^3 + \frac{1}{x^3} = 110$

Summary

By first deriving $x + 1/x = 5$ from the given equation $x^2 - 5x + 1 = 0$ and then applying the sum of cubes identity, we found the value of $x^3 + 1/x^3$ to be 110.