Question

Trayvon was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?
zeros x=-5 mutiplicity 2, x=-10 mutiplicity 5

Answer 1

To construct a polynomial function based on the given zeros and their multiplicities, we can use the fact that if a polynomial has a zero at \( x = r \) with multiplicity \( m \), it can be represented as \( (x - r)^m \).

Given the zeros and their multiplicities:
- Zero at \( x = -5 \) with multiplicity 2: This contributes \( (x + 5)^2 \).
- Zero at \( x = -10 \) with multiplicity 5: This contributes \( (x + 10)^5 \).

Now, we can combine these contributions to write the polynomial function:

\[
f(x) = (x + 5)^2 (x + 10)^5
\]

To express this polynomial in standard form (though not strictly necessary for identifying it), one could expand this product, but it's not always necessary unless specifically requested.

Thus, the polynomial function Trayvon should write is:

\[
f(x) = (x + 5)^2 (x + 10)^5
\]