Answer :
To construction of a polynomial function with specific properties, we break down the given information step by step to construct the polynomial function.
1. The polynomial function is fifth degree, meaning it has a highest degree of 5.
2. One of the zeros is 5, and it has a multiplicity of 4. This means that the polynomial function has a repeated root of 5 four times.
3. The only other zero is -4.
4. The leading coefficient is 3.
To construct the polynomial function, we can start by writing the factored form of the function:
[tex]f(x) = 3(x-5)^4 (x+4)[/tex]
To find the expanded form, we can multiply the factors:
[tex]f(x) = 3(x^4 - 20x^3 + 150x^2 - 500x + 625) (x+4)[/tex]
Next, we can distribute the 3 to each term inside the parentheses:
[tex]f(x) = 3(x^5 + 4x^4 - 20x^4 - 80x^3 + 150x^3 + 600x^2 - 500x^2 - 2000x + 625x + 2500)[/tex]
Simplifying further, we combine like terms:
[tex]f(x) = 3(x^5 - 16x^4 + 70x^3 + 100x^2 - 1375x + 2500)[/tex]
Finally, we can reorder the terms in descending order of their degrees:
[tex]f(x) = 3x^5 - 48x^4 + 210x^3 + 300x^2 - 4125x + 7500[/tex]
Therefore, the polynomial function that satisfies the given conditions is:
[tex]f(x) = 3x^5 - 48x^4 + 210x^3 + 300x^2 - 4125x + 7500[/tex]
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