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Let's address the questions based on the polynomial function and its characteristics.

### Question 12:
The function given is:
\[
y = -x^4 + 3x^2 + 5
\]
To determine how many **x-intercepts** this function has, we need to set \( y = 0 \) and solve for \( x \):

\[
0 = -x^4 + 3x^2 + 5
\]

Rearranging the equation:
\[
x^4 - 3x^2 - 5 = 0
\]

Let \( z = x^2 \). Substituting gives:
\[
z^2 - 3z - 5 = 0
\]
We solve this quadratic equation using the quadratic formula:
\[
z = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-5)}}{2(1)} = \frac{3 \pm \sqrt{9 + 20}}{2} = \frac{3 \pm \sqrt{29}}{2}
\]

Since \( \sqrt{29} \) is a positive real number, we get two real solutions for \( z \). However, \( z = x^2 \), so each solution for \( z \) will yield two values of \( x \) (since \( x = \pm \sqrt{z} \)).

Both solutions for \( z \) are positive, indicating real solutions for \( x \). Therefore, the function has **4 x-intercepts**.

**Answer for Question 12: 4**

---

### Question 13:
For an **even polynomial function with a positive leading coefficient**, the minimum number of x-intercepts is **0**. This is because the graph of an even-degree polynomial (like \(x^4\)) opens either upwards or downwards. In some cases, it may not intersect the x-axis at all.

**Answer for Question 13: 0**

---

### Question 14:
For an **even polynomial function with a positive leading coefficient**, the maximum number of x-intercepts is determined by the degree of the polynomial. Since the highest degree of the polynomial is 4 (even degree), the function can have a maximum of 4 real x-intercepts.

**Answer for Question 14: 4**

---

Would you like any further explanations or have more questions? Here are some related questions to explore further:

1. How do the x-intercepts of odd-degree polynomials differ from even-degree ones?
2. How do you determine the number of turning points for a polynomial function?
3. What is the role of the leading coefficient in determining the direction of the graph of a polynomial?
4. How do you solve a quadratic equation using the quadratic formula step by step?
5. What happens when a polynomial function has repeated roots?

**Tip**: When solving polynomial equations, always check if factoring can simplify the process before using more complex methods like the quadratic formula.

The function y = -x^4 + 3x^2 + 5 has _____ x-intercepts. For an even polynomial function with a positive leading coefficient, what is the minimum and maximum number of x-intercepts?

Explore how to solve the polynomial function y = -x^4 + 3x^2 + 5 to determine its x-intercepts. Understand the behavior of even-degree polynomials with positive leading coefficients, and learn the minimum and maximum possible x-intercepts. Suitable for students in Grades 10-12.

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