Answer:
To find the factored form of the polynomial function, we start by using the zeros to write out the factors of the polynomial. If the zeros are a, b, c, d, then the factors are (x-a), (x-b), (x-c), and (x-d).
In this case, the zeros are -1, -3, 3, and 1, so the factors are (x+1), (x+3), (x-3), and (x-1). Multiplying these factors together gives the polynomial function:
f(x) = (x+1)(x+3)(x-3)(x-1)
To simplify this expression, we can use the distributive property to expand the factors:
f(x) = (x+1)(x^2 - 9)(x-1)
Next, we can multiply the remaining factors using the distributive property and combining like terms:
f(x) = (x^3 - 8x - 9)(x-1)
Finally, we can use the distributive property again to expand the remaining factor:
f(x) = x^4 - 9x^2 + 8x + 9x^3 - 8x - 9
Simplifying this expression gives the factored form of the polynomial function:
f(x) = (x+1)(x+3)(x-3)(x-1) = x^4 - 9x^2 + 8x - 9x^3 - 8x - 9.