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Create a polynomial, f(x), with real coefficients having a degree of 3 and zeros -2 and 3+i.

a. f(x) = (x + 2)(x - 3 - i)(x - 3 + i)
b. f(x) = (x - 2)(x + 3 - i)(x + 3 + i)
c. f(x) = (x + 2)(x + 3 - i)(x + 3 + i)
d. f(x) = (x - 2)(x - 3 - i)(x - 3 + i)

User Ivodvb
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Final answer:

The polynomial with a degree of 3 and zeros -2, 3+i is f(x) = (x + 2)(x - 3 - i)(x - 3 + i). This is correct because complex zeros come in conjugate pairs, and the polynomial is constructed accordingly.

Step-by-step explanation:

To create a polynomial, f(x), with a degree of 3 and given zeros -2 and 3+i, we need to consider that the zeros of a polynomial with real coefficients that are complex must come in conjugate pairs. This means that if 3+i is a zero, then its conjugate 3-i is also a zero. Using this information, we can construct the polynomial.

The correct polynomial with the given zeros -2, 3+i, and 3-i is expressed as f(x) = (x + 2)(x - 3 - i)(x - 3 + i). To confirm, we can multiply out the factors:

(x + 2)(x - 3 - i)(x - 3 + i) = (x + 2)[(x - 3) - i][(x - 3) + i] = (x + 2)((x - 3)^2 - (i)^2) = (x + 2)(x^2 - 6x + 9 + 1) = (x + 2)(x^2 - 6x + 10)

Expanding and simplifying these expressions, we find:

f(x) = x^3 - 4x^2 + 2x + 20

This confirms that the correct answer is (a), which is the polynomial having a degree of 3 and zeros -2, 3+i, and 3-i.

User Cleary
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