Answer:
To find the polynomial function in standard form with the given zeros, we need to consider the multiplicity of each zero.
The zero 1 has a multiplicity of 2, which means it appears twice as a root of the polynomial. Similarly, the zero -2 has a multiplicity of 3, meaning it appears three times as a root.
To construct the polynomial function, we can use the following steps:
1. For the zero 1 with a multiplicity of 2, we know that (x - 1) should appear twice as a factor in the polynomial. Therefore, we have (x - 1)^2 = x^2 - 2x + 1.
2. For the zero -2 with a multiplicity of 3, we know that (x + 2) should appear three times as a factor in the polynomial. Therefore, we have (x + 2)^3 = x^3 + 6x^2 + 12x + 8.
Now, we multiply these factors together to obtain the polynomial function:
f(x) = (x^2 - 2x + 1)(x^3 + 6x^2 + 12x + 8)
Simplifying this expression by expanding and combining like terms, we get:
f(x) = x^5 + 4x^4 + x^3 - 10x^2 - 4x + 8
Comparing this with the answer choices provided, we can see that the correct polynomial function in standard form is option C:
f(x) = x^4 + 4x^2 + x^3 - 10x^2 - 4x + 8
Explanation:
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