The graph of a polynomial function has zeros of 0 (multiplicity 2), 5 (multiplicity 3) and -1 (multiplicity 4). Write a function in factored form (least too greatest by exponents) that could represent this function.

Answer 1

To write a polynomial function in factored form based on the given zeros and their multiplicities, we use the following information:

• The zero at \(0\) has a multiplicity of \(2\): This contributes the factor \(x^2\).
• The zero at \(5\) has a multiplicity of \(3\): This contributes the factor \((x - 5)^3\).
• The zero at \(-1\) has a multiplicity of \(4\): This contributes the factor \((x + 1)^4\).

Now, we can combine these factors to write the polynomial function in factored form:

\[ f(x) = x^2 (x - 5)^3 (x + 1)^4 \]

This polynomial is expressed in factored form, taking into account the multiplicities and the order in which factors were included. Hence, the final form is:

\[ f(x) = x^2 (x - 5)^3 (x + 1)^4 \]