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The given equation is \( y = (x^2 + 1)(x^2 - 1)(2x + 1) \), and you're tasked with modifying it to meet the following conditions:

1. Leading term is \( 2x^5 \)
2. Has two non-real complex roots
3. Has a real root at \( x = -\frac{1}{2} \)

Let's break down these conditions:

### (1) Leading term is \( 2x^5 \):
- The degree of the polynomial should be 5, and the coefficient of the highest-degree term should be 2. 

In the current equation:
\[
y = (x^2 + 1)(x^2 - 1)(2x + 1)
\]
- The degree is 5 because we have two quadratic factors (each contributing \(x^2\)) and one linear factor.
- To get a leading coefficient of 2, we can keep the factor \( 2x + 1 \), as this already ensures that the leading coefficient is 2.

### (2) Has two non-real complex roots:
- Complex roots typically come from quadratic factors with no real roots, such as \( x^2 + 1 \). In the current equation, \( (x^2 + 1) \) provides complex roots \( x = i \) and \( x = -i \), so we can keep this factor unchanged.

### (3) Has a real root at \( x = -\frac{1}{2} \):
- A real root at \( x = -\frac{1}{2} \) comes from the factor \( 2x + 1 \) because:
\[
2x + 1 = 0 \implies x = -\frac{1}{2}
\]
This condition is already satisfied by the factor \( 2x + 1 \).

### Modified Equation:
We can change \( (x^2 - 1) \) to \( (x - 1)^2 \), which still keeps the degree at 5 and adds a double root at \( x = 1 \). The new equation will then be:
\[
y = (x + 1)(x - 1)^2(2x + 1)
\]
This meets all the given requirements:
- The leading term is \( 2x^5 \).
- The quadratic \( (x + 1) \) has complex roots.
- The factor \( (2x + 1) \) ensures a real root at \( x = -\frac{1}{2} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the leading term influence the overall behavior of the polynomial?
2. What are the steps to factor polynomials with complex roots?
3. How do real and complex roots affect the graph of a polynomial?
4. Can a polynomial have both real and complex roots?
5. What does it mean for a root to have multiplicity (e.g., \( (x - 1)^2 \))?

**Tip:** When modifying equations, always keep track of the degree and leading coefficient to ensure the function behaves as desired.

Change one or more numbers in the equation to create another polynomial that meets the requirements: 1. Leading term is 2x^5, 2. Has two non-real complex roots, 3. Has a real root at x = -1/2.

Learn how to modify a polynomial equation to meet the requirements: a leading term of 2x^5, two non-real complex roots, and a real root at x = -1/2. This problem covers concepts of polynomials, complex roots, and real root factors, suitable for high school students.

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