Math Problem Statement
Find a polynomial function that has the given zeros. (There are many correct answers.)
0, 1, 5
Solution
To find a polynomial function with the given zeros , , and , we can write the polynomial as a product of factors corresponding to these zeros.
The factors will be of the form . So, the factors for the given zeros are:
Now, multiply these factors together to get the polynomial:
Now expand this expression:
- Multiply the first two factors:
- Now multiply by the third factor :
Thus, one possible polynomial function is:
Final Answer:
This is a polynomial with the given zeros , , and . There are infinitely many such polynomials because you can multiply this polynomial by any constant, and the resulting function will still have the same zeros.
Would you like a more detailed explanation of the process? Here are some related questions for further practice:
- What happens to the polynomial if we multiply it by a constant, say 2?
- How do you determine the degree of the polynomial from the number of zeros?
- Can a polynomial with real coefficients have complex zeros?
- How would you verify that these zeros satisfy the polynomial?
- How do you find a polynomial if given complex or repeated zeros?
Tip: A polynomial of degree has at most zeros, including multiplicity.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
Factoring
Formulas
Factoring polynomial from zeros: f(x) = (x - zero1)(x - zero2)...
Theorems
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12