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img.show()I've displayed the image to analyze its content. The document seems to describe questions on handling polynomial functions, specifically focusing on dealing with complex and irrational zeros, and expanding polynomials.

Let me summarize the questions and provide answers:

1. **Handling complex or irrational zeros:**
   - When you have complex or irrational zeros like \( 2 + 2i \) or \( 5\sqrt{5} \), you need to account for **conjugate pairs**. This means if \( 2 + 2i \) is a zero, \( 2 - 2i \) must also be a zero (and vice versa). Similarly, for an irrational zero like \( 5\sqrt{5} \), its conjugate, \( -5\sqrt{5} \), is also a zero.
   - **Factoring**: If the complex zeros are given, you can create factors from them. For example, if \( 2 + 2i \) is a zero, the factor would be \( (x - (2 + 2i)) \), and for its conjugate, it would be \( (x - (2 - 2i)) \). Multiplying these conjugate factors results in a quadratic factor with real coefficients.

2. **Leading coefficient other than 1:**
   - If the problem gives a leading coefficient other than 1, you should **scale the entire polynomial** by the leading coefficient. This means, after expanding and simplifying the factors, multiply the whole polynomial by the given leading coefficient to match the required form.
   - For example, if the polynomial has a factor like \( (x-2) \) and the leading coefficient is 3, you would multiply the entire polynomial by 3 after expansion to adjust the coefficient.

3. **Expanding and simplifying the polynomial:**
   - When expanding, it’s easy to make mistakes distributing or combining like terms. Always be systematic:
     1. First expand two terms at a time.
     2. Then, gradually combine like terms to avoid errors.
     3. Finally, double-check each step, especially when handling larger polynomials.

Would you like more details or assistance with specific problems on these topics? Here are a few related questions you might find helpful:
1. How do you factor polynomials with complex conjugates in general?
2. Can you provide an example of expanding a polynomial with irrational zeros?
3. What is the effect of a leading coefficient of -1 on the expanded form of a polynomial?
4. How do you simplify a large polynomial after expanding many terms?
5. Can you explain the conjugate rule with specific examples and its importance in polynomial functions?

Tip: Always remember that for any complex or irrational root, its conjugate must also be a root in order for the polynomial to have real coefficients.

I’m having trouble writing polynomial functions when given a set of zeros. I understand that if a polynomial has zeros like x=2, -3, x = 2, -3, x=2, -3, and 444, then the factors should be (x−2)(x+3)(x−4)(x−2)(x+3)(x−4). However, I get confused when:

1. There are complex or irrational zeros (e.g., 2+2i + i2+i or 5√5). How do I account for their conjugates?

2. The problem gives a leading coefficient other than 1. How does that affect the function?

3. I need to expand and simplify the factors into standard form. I sometimes make mistakes distributing or combining like terms.

If anyone has tips, especially on handling complex/irrational zeros or expanding polynomials efficiently, I’d appreciate the help.

Learn how to write polynomial functions from given zeros, including handling complex and irrational roots, dealing with leading coefficients, and expanding polynomials efficiently. Understand the complex conjugate root theorem and polynomial expansion techniques with step-by-step explanations.

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