Let (a - 1) (x² + 3½ x + 1 )² z (a+ 1)(x⁴ - x² + 1) less than or equal to 0 for all x , then 1) sum of integral values a is

Given the inequality:

\[
(a - 1) \cdot \left( x^2 + \frac{7}{2} x + 1 \right)^2 \cdot z \cdot (a + 1) \cdot (x^4 - x^2 + 1) \leq 0
\]

where \( z \) is a function of \( x \), we need to determine the sum of all possible integral values of \( a \) such that the inequality holds for all \( x \).

### Step 1: Analyze the factors

1. **Factor 1:** \( (a - 1) \)
2. **Factor 2:** \( \left( x^2 + \frac{7}{2} x + 1 \right)^2 \) - This expression is always non-negative since it's a square.
3. **Factor 3:** \( z \)
4. **Factor 4:** \( (a + 1) \)
5. **Factor 5:** \( x^4 - x^2 + 1 \) - This expression is always positive for all \( x \) because \( x^4 - x^2 + 1 > 0 \).

### Step 2: Condition for Inequality

For the inequality to hold for all \( x \), the product must be non-positive. The non-positive condition can only be fulfilled if \( (a - 1)(a + 1) \) is non-positive (i.e., negative or zero), given that the other terms do not change sign (since they are non-negative or positive for all \( x \)).

### Step 3: Analyze \( (a - 1)(a + 1) \leq 0 \)

This condition holds when:

\[
a^2 - 1 \leq 0
\]

Solving:

\[
(a - 1)(a + 1) \leq 0 \implies -1 \leq a \leq 1
\]

So, the integral values \( a \) can take are \( -1, 0, 1 \).

### Step 4: Sum of Integral Values

The sum of these integral values is:

\[
-1 + 0 + 1 = 0
\]

### Final Answer

The sum of the integral values of \( a \) is **0**.

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Explore how to find the sum of integral values of a in the inequality (a - 1)(x² + 3½x + 1)² z (a + 1)(x⁴ - x² + 1) ≤ 0. Understand the mathematical conditions and steps required to solve this advanced algebraic inequality. Suitable for advanced high school students.

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