Question
The graph of a polynomial function has zeros of 0 (multiplicity 2), 1 (multiplicity 2), and 5/2 (multiplicity 2). Write a function in standard form that could represent this function.
Ask by Watkins Riley. in the United States
Mar 25,2025
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Answer
A possible polynomial function with zeros at 0 (multiplicity 2), 1 (multiplicity 2), and 5/2 (multiplicity 2) is:
Solution
The zeros given are
where
Thus, we have
We now expand the expression step by step.
-
Write the factors
and explicitly: -
Multiply
and : -
Combine like terms:
-
The
term: . -
The
terms: . -
The
terms: . Write . Then, -
The
terms: . Write . Then, -
The constant term:
.
So, the product becomes: -
-
Now multiply this result by
:
Thus, one possible function in standard form that has the given zeros is
This is the answer.
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Simplify this solution Beyond the Answer
To find a polynomial function that has the given zeros with their respective multiplicities, we can express it in factored form first. Since the zero at 0 has a multiplicity of 2, we can write
Now, putting it all together, the polynomial in factored form is:
where
The polynomial function in standard form is:
After expanding, you’d obtain:
Further simplification will yield the complete polynomial function.
This yields a function with the desired properties and showcases the neat relationship between zeros and multiplicities!
