The given expression is \( P(-x) = 3(-x)^4 - 7(-x)^2 + 9 \). Let's simplify it step by step.

### Step-by-Step Simplification:

1. **Simplify the powers of \(-x\):**
   - \( (-x)^4 = x^4 \), since raising a negative number to an even power results in a positive number.
   - \( (-x)^2 = x^2 \), for the same reason as above.

So, the expression becomes:
\[
P(-x) = 3x^4 - 7x^2 + 9
\]

This is the same as the original function \( P(x) = 3x^4 - 7x^2 + 9 \).

### Conclusion:
\[
P(-x) = P(x)
\]
This shows that the function is **even**, meaning it satisfies the property \( P(-x) = P(x) \) for all \( x \).

Would you like any further details or have any questions about this?

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Here are five related questions you might find interesting:
1. What is the definition of an even function, and how does it differ from an odd function?
2. How can you determine the symmetry of a polynomial function graphically?
3. Can you find the degree and leading coefficient of \( P(x) = 3x^4 - 7x^2 + 9 \)?
4. How would \( P(x) \) change if the powers of \( x \) were odd instead of even?
5. How would you approach simplifying \( P(-x) \) if more complex functions like trigonometric ones were involved?

**Tip:** Even functions are symmetric about the y-axis, while odd functions exhibit rotational symmetry about the origin.

Explore the process of simplifying the polynomial function P(-x) = 3(-x)^4 - 7(-x)^2 + 9 and determine its evenness. Learn key algebraic concepts like function symmetry and simplifying polynomial powers. Suitable for Grades 10-12 students studying polynomial functions and symmetry.

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