To solve this problem, let’s analyze the provided information and simplify the expression step-by-step.

Given Information

We have complex numbers $x$, $y$, and $z$ such that:

1. $x + y + z = 2$,
2. $x^2 + y^2 + z^2 = 3$,
3. $xyz = 4$.

We are asked to evaluate: $\frac{1}{xy + z - 1} + \frac{1}{yz + x - 1} + \frac{1}{zx + y - 1}.$

Solution Steps

Step 1: Express $xy + z$, $yz + x$, and $zx + y$ in terms of symmetric sums

We will use the identities involving symmetric sums of $x$, $y$, and $z$.

Let:

• $S_1 = x + y + z = 2$,
• $S_2 = xy + yz + zx$,
• $S_3 = xyz = 4$.

Using the identity for the sum of squares: $x^2 + y^2 + z^2 = S_1^2 - 2S_2.$ Substituting the values we know: $3 = 2^2 - 2S_2 \Rightarrow 3 = 4 - 2S_2 \Rightarrow S_2 = \frac{1}{2}.$

So, we have:

• $S_1 = 2$,
• $S_2 = \frac{1}{2}$,
• $S_3 = 4$.

Step 2: Simplify each term in the expression

Now, let’s substitute these values into the expression we need to evaluate: $\frac{1}{xy + z - 1} + \frac{1}{yz + x - 1} + \frac{1}{zx + y - 1}.$

Each term has the form $\frac{1}{S_2 - 1}$, so we substitute $S_2 = \frac{1}{2}$: $S_2 - 1 = \frac{1}{2} - 1 = -\frac{1}{2}.$

Thus, each fraction becomes: $\frac{1}{-\frac{1}{2}} = -2.$

Step 3: Add the terms

Since there are three terms, each equal to $-2$: $-2 + (-2) + (-2) = -6.$

Final Answer

The value of the expression is: $\boxed{-6}.$

Would you like a further breakdown of any step, or do you have any questions?

Related Questions

1. How would the result change if $xyz$ had a different value?
2. Can we generalize this approach for polynomials of higher degrees?
3. What if the expression to evaluate involved different combinations of terms?
4. How would this problem be affected if we were working with real numbers instead of complex?
5. What other identities could be useful for solving similar symmetric sum problems?

Tip

Using symmetric sums is a powerful technique for solving problems involving sums and products of roots without knowing the exact values of each root.