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Answer :
To find all the zeros of the polynomial [tex]\( f(x) = 2x^5 + 15x^4 + 60x^3 + 30x^2 - 182x + 75 \)[/tex], given that two of its zeros are [tex]\( -3 - 4i \)[/tex] and [tex]\( \frac{1}{2} \)[/tex], we can use the following steps:
1. Use the Conjugate Root Theorem:
Since [tex]\( -3 - 4i \)[/tex] is a zero, and the coefficients of the polynomial are real numbers, its complex conjugate [tex]\( -3 + 4i \)[/tex] must also be a zero. So, we have three zeros: [tex]\( -3 - 4i \)[/tex], [tex]\( -3 + 4i \)[/tex], and [tex]\( \frac{1}{2} \)[/tex].
2. Factor Setup for Known Zeros:
The polynomial can be factored to include these known zeros:
[tex]\[
f(x) = (x - (-3 - 4i))(x - (-3 + 4i))(x - \frac{1}{2})g(x)
\][/tex]
Where [tex]\( g(x) \)[/tex] is a reduced polynomial after factoring out the known zeros.
3. Factor the Complex Conjugate Pair:
Multiply the factors associated with the complex zeros:
[tex]\[
(x - (-3 - 4i))(x - (-3 + 4i)) = ((x + 3) - 4i)((x + 3) + 4i) = (x + 3)^2 + 16
\][/tex]
Resulting in:
[tex]\[
(x + 3)^2 + 16 = x^2 + 6x + 9 + 16 = x^2 + 6x + 25
\][/tex]
4. Determine Remaining Zeros:
The remaining polynomial [tex]\( g(x) \)[/tex] can be found by dividing the original polynomial by [tex]\((x^2 + 6x + 25)(x - \frac{1}{2})\)[/tex].
5. Check for Remaining Zeros:
By solving for [tex]\( g(x) \)[/tex], we find two additional zeros, which alongside the previously determined zeros, make up the complete set:
- [tex]\( -3 \)[/tex]
- [tex]\( \frac{1}{2} \)[/tex]
- [tex]\( 1 \)[/tex]
- [tex]\(-3 - 4i\)[/tex]
- [tex]\(-3 + 4i\)[/tex]
Thus, the zeros of the polynomial [tex]\( f(x) \)[/tex] are:
[tex]\[
-3, \frac{1}{2}, 1, -3 - 4i, -3 + 4i
\][/tex]
1. Use the Conjugate Root Theorem:
Since [tex]\( -3 - 4i \)[/tex] is a zero, and the coefficients of the polynomial are real numbers, its complex conjugate [tex]\( -3 + 4i \)[/tex] must also be a zero. So, we have three zeros: [tex]\( -3 - 4i \)[/tex], [tex]\( -3 + 4i \)[/tex], and [tex]\( \frac{1}{2} \)[/tex].
2. Factor Setup for Known Zeros:
The polynomial can be factored to include these known zeros:
[tex]\[
f(x) = (x - (-3 - 4i))(x - (-3 + 4i))(x - \frac{1}{2})g(x)
\][/tex]
Where [tex]\( g(x) \)[/tex] is a reduced polynomial after factoring out the known zeros.
3. Factor the Complex Conjugate Pair:
Multiply the factors associated with the complex zeros:
[tex]\[
(x - (-3 - 4i))(x - (-3 + 4i)) = ((x + 3) - 4i)((x + 3) + 4i) = (x + 3)^2 + 16
\][/tex]
Resulting in:
[tex]\[
(x + 3)^2 + 16 = x^2 + 6x + 9 + 16 = x^2 + 6x + 25
\][/tex]
4. Determine Remaining Zeros:
The remaining polynomial [tex]\( g(x) \)[/tex] can be found by dividing the original polynomial by [tex]\((x^2 + 6x + 25)(x - \frac{1}{2})\)[/tex].
5. Check for Remaining Zeros:
By solving for [tex]\( g(x) \)[/tex], we find two additional zeros, which alongside the previously determined zeros, make up the complete set:
- [tex]\( -3 \)[/tex]
- [tex]\( \frac{1}{2} \)[/tex]
- [tex]\( 1 \)[/tex]
- [tex]\(-3 - 4i\)[/tex]
- [tex]\(-3 + 4i\)[/tex]
Thus, the zeros of the polynomial [tex]\( f(x) \)[/tex] are:
[tex]\[
-3, \frac{1}{2}, 1, -3 - 4i, -3 + 4i
\][/tex]
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