If x, y and z are the three real roots of the cubic equation t^3 − 7t − 6 = 0, what is the value of x + y + z?

Introduction / Context:
This algebra question asks for the sum of the roots of a cubic polynomial using the standard relationship between coefficients and roots. Rather than solving the cubic explicitly, you can use Vieta's formulas to relate the sum and products of roots directly to the coefficients of the polynomial. This is an essential technique in polynomial algebra and aptitude exams.

Given Data / Assumptions:

• The cubic equation is t^3 − 7t − 6 = 0.
• Its real roots are x, y and z.
• We need the value of x + y + z.

Concept / Approach:
For a general cubic equation of the form t^3 + pt^2 + qt + r = 0 with roots r₁, r₂ and r₃, Vieta's formulas give: r₁ + r₂ + r₃ = −p. We identify p, q and r from the given polynomial and apply this formula directly, without solving for the individual roots.

Step-by-Step Solution:
Write the equation in standard form: t^3 − 7t − 6 = 0. Compare with t^3 + pt^2 + qt + r = 0. Here, the coefficient of t^3 is 1, the coefficient of t^2 is 0, the coefficient of t is −7 and the constant term is −6. Thus p = 0, q = −7 and r = −6. If the roots are x, y, z, then by Vieta's formulas: x + y + z = −p = −0 = 0.
Verification / Alternative check:
We can factor the polynomial to confirm. Try small integer values for t. At t = 3: 27 − 21 − 6 = 0, so t = 3 is a root. Divide t^3 − 7t − 6 by (t − 3) to obtain the quadratic factor t^2 + 3t + 2. This quadratic has roots −1 and −2. Therefore the three roots are 3, −1 and −2. Their sum is 3 + (−1) + (−2) = 0, which agrees with Vieta's formula.

Why Other Options Are Wrong:
The options 3 or 6 might come from wrongly assuming that the sum equals either the coefficient of t or the constant term. The expression 3a or a does not have any meaning here because the variable in the polynomial is t, and there is no parameter a in this equation. Only the Vieta-based value 0 is correct.

Common Pitfalls:
A common mistake is to mix up the signs in Vieta's formulas or to misidentify the coefficients. Another potential error is trying to solve the cubic completely when only the sum of the roots is required. Remember that for t^3 + pt^2 + qt + r = 0, the sum of the roots is simply −p.

Final Answer:
The value of x + y + z is 0.