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Given that-4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. fQ) = x* + x3 + 10x + 16x - 96

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Final answer:

To factor the given polynomial function completely, we use the Conjugate Roots Theorem. The factors of the polynomial are (x + 4i), (x - 4i), and (x - 6).

Step-by-step explanation:

To factor the given polynomial function completely, we first need to find its zeros. Since -4i is given as a zero, we know that its conjugate, 4i, must also be a zero. Therefore, the factors of the polynomial are (x + 4i) and (x - 4i). Now, we can divide the polynomial by these factors using polynomial long division or synthetic division to find the remaining factor. The resulting factor is (x - 6).

So, the completely factored form of the polynomial function is f(x) = (x + 4i)(x - 4i)(x - 6).

User Bisi
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Final answer:

To factor the given polynomial function, use the Conjugate Roots Theorem and divide the polynomial by the factors obtained from the zeros. The fully factored form of the polynomial function is
(x - 4i)(x + 4i)(x^2 + 6x - 6).

Step-by-step explanation:

To factor the polynomial function
f(x) = x^4 + x^3 + 10x^2 + 16x - 96, we start by using the Conjugate Roots Theorem.

Since -4i is a zero, we know that 4i is also a zero.

So the factors are (x + 4i) and (x - 4i). We can write them as (x - 4i)(x + 4i).

Next, we can divide the polynomial f(x) by (x - 4i)(x + 4i) to obtain the remaining factor.

Using long division or synthetic division, we find that the remaining factor is
x^2 + 6x - 6.

Therefore, the fully factored form of the polynomial function is
f(x) = (x - 4i)(x + 4i)(x^2 + 6x - 6).

User Smbear
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