Question
Form a polynomial whose real zeros and degree are given.
Zeros:
; degree: 3
Zeros:
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:
-
Zeros and Corresponding Factors:
- Zero at
leads to the factor - Zero at
leads to the factor - Zero at
leads to the factor
- Zero at
-
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). -
Example Polynomial (with
): -
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.