Question
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Form a polynomial whose real zeros and degree are given.
Zeros: ; degree: 3

Ask by Bob Malone. in the United States
Jan 29,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

A polynomial with zeros at , , and and degree 3 is:
Or, expanded:

Solution

To form a polynomial of degree 3 with real zeros at , , and , you can use the factored form of the polynomial. Each zero corresponds to a factor of the polynomial:
  1. Zeros and Corresponding Factors:
    • Zero at leads to the factor
    • Zero at leads to the factor
    • Zero at leads to the factor
  2. Forming the Polynomial:
    Combine these factors to form the polynomial:
    Here, is a non-zero constant that can be chosen based on specific requirements (often set to 1 for simplicity).
  3. Example Polynomial (with ):
  4. Expanded Form (Optional):
    If you prefer the expanded form, multiply the factors out:
Final Polynomial:
Or, in expanded form:

Answered by UpStudy AI and reviewed by a Professional Tutor

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Bonus Knowledge

To form a polynomial with the given zeros , , and , we can express the polynomial as a product of its factors. The polynomial can be written as:
where is a non-zero constant that adjusts the leading coefficient. Since the degree is 3, we can simply set for simplicity. Thus, the polynomial becomes:
Now, expanding this gives us:
Now, will expand further:
So, the polynomial is:
This polynomial has the desired zeros and is of degree 3.

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