XWhich polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 – i) with multiplicity 1?a. f(x) = (x + 7) (x – i) (x + 5) (x + i)
b. f(x) = (x – 7) (x – i) (x – 5) (x + i)
c. f(x) = (x – (7 – i)) (x – (5 + i)) (x – (7 + i)) (x – (5 – i))
d. f(x) = (x + (7 – i)) (x + (5 + i)) (x + (7 + i)) (x + (5 – i))

Answers

Answer 1

Answer: Hello,

If ALL coefficients of x are reals!, (question is wrong)
if $z_(0)\ is\ a\ root\ \overline{z_(0)}\ is\ also\ a\ root$

Thus ANSWER C f(x) = (x – (7 – i)) (x – (5 + i)) (x – (7 + i)) (x – (5 – i))

Answer 2

Answer:

Final answer:

The polynomial function with a leading coefficient of 1 and roots (7 + i) and (5 – i) with multiplicity 1 is f(x) = (x + 7) (x – i) (x + 5) (x + i).

Explanation:

The polynomial function with a leading coefficient of 1 and roots (7 + i) and (5 – i) with multiplicity 1 is option a. f(x) = (x + 7) (x – i) (x + 5) (x + i). To understand why this is the correct answer, we first need to know that complex roots always appear in conjugate pairs, which means that if a + bi is a root, then a - bi is also a root. The given roots are (7 + i) and (5 – i), so the conjugate pairs are (7 – i) and (5 + i).

Therefore, the correct polynomial is obtained by multiplying the factors (x – (7 + i)), (x – (7 – i)), (x – (5 + i)), and (x – (5 – i)). This gives us f(x) = (x + 7) (x – i) (x + 5) (x + i), which is option a.

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