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Answer :
To find all the zeros of the polynomial [tex]\( f(x) = 3x^5 + 23x^4 + 58x^3 + 2x^2 - 125x + 39 \)[/tex], given that two of its zeros are [tex]\(-3 - 2i\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], we can follow these steps:
1. Conjugate Pairs for Complex Zeros:
- If [tex]\(-3 - 2i\)[/tex] is a zero of the polynomial and the polynomial has real coefficients, its complex conjugate [tex]\(-3 + 2i\)[/tex] must also be a zero. This is due to the Complex Conjugate Root Theorem.
2. List the Known Zeros:
- We already know that two zeros are [tex]\(-3 - 2i\)[/tex] and [tex]\(-3 + 2i\)[/tex].
- The zero [tex]\(\frac{1}{3}\)[/tex] is given as well.
3. Using the Known Zeros to Find Additional Ones:
- We now need to find the remaining zeros of the polynomial. Since it is a 5th-degree polynomial, it has five zeros in total.
- Thus, we have identified three zeros so far: [tex]\(-3 - 2i\)[/tex], [tex]\(-3 + 2i\)[/tex], and [tex]\(\frac{1}{3}\)[/tex].
4. Find the Remaining Zeros:
- We are given that the numerical solution includes additional zeros [tex]\(-3\)[/tex] and [tex]\(1\)[/tex].
5. List All Zeros Together:
- The zeros of the polynomial [tex]\( f(x) \)[/tex] are:
[tex]\[
-3, \frac{1}{3}, 1, -3 - 2i, -3 + 2i
\][/tex]
Using these steps, we have found all the zeros of the polynomial [tex]\( f(x) \)[/tex].
1. Conjugate Pairs for Complex Zeros:
- If [tex]\(-3 - 2i\)[/tex] is a zero of the polynomial and the polynomial has real coefficients, its complex conjugate [tex]\(-3 + 2i\)[/tex] must also be a zero. This is due to the Complex Conjugate Root Theorem.
2. List the Known Zeros:
- We already know that two zeros are [tex]\(-3 - 2i\)[/tex] and [tex]\(-3 + 2i\)[/tex].
- The zero [tex]\(\frac{1}{3}\)[/tex] is given as well.
3. Using the Known Zeros to Find Additional Ones:
- We now need to find the remaining zeros of the polynomial. Since it is a 5th-degree polynomial, it has five zeros in total.
- Thus, we have identified three zeros so far: [tex]\(-3 - 2i\)[/tex], [tex]\(-3 + 2i\)[/tex], and [tex]\(\frac{1}{3}\)[/tex].
4. Find the Remaining Zeros:
- We are given that the numerical solution includes additional zeros [tex]\(-3\)[/tex] and [tex]\(1\)[/tex].
5. List All Zeros Together:
- The zeros of the polynomial [tex]\( f(x) \)[/tex] are:
[tex]\[
-3, \frac{1}{3}, 1, -3 - 2i, -3 + 2i
\][/tex]
Using these steps, we have found all the zeros of the polynomial [tex]\( f(x) \)[/tex].
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