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Answer :
To find the complex zeros of the polynomial function [tex]\( f(x) = -3x^5 + 35x^4 - 129x^3 + 139x^2 - 16x - 26 \)[/tex] and write the polynomial in completely factored form, follow these steps:
1. Identify the Zeros:
Based on the results, the complex zeros of the polynomial are:
- [tex]\( x = -\frac{1}{3} \)[/tex]
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = 5 - i \)[/tex]
- [tex]\( x = 5 + i \)[/tex]
2. Understand the Multiplicity:
Each of these zeros appears once, so they have a multiplicity of 1.
3. Write the Polynomial in Factor Form:
Knowing the zeros, we can express the polynomial in its completely factored form:
[tex]\[
f(x) = -3 \left( x + \frac{1}{3} \right) (x - 1) (x - (5 - i)) (x - (5 + i))
\][/tex]
4. Simplify the Factored Form:
To simplify, start by adjusting the factor [tex]\( x + \frac{1}{3} \)[/tex] to avoid fractions:
[tex]\[
-3\left(x + \frac{1}{3}\right) = (-3)(x + \frac{1}{3}) = -3x - 1
\][/tex]
The final completely factored form is:
[tex]\[
f(x) = (-3x - 1)(x - 1)(x - (5 - i))(x - (5 + i))
\][/tex]
This gives us the polynomial expressed in factored form showcasing all its complex zeros.
1. Identify the Zeros:
Based on the results, the complex zeros of the polynomial are:
- [tex]\( x = -\frac{1}{3} \)[/tex]
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = 5 - i \)[/tex]
- [tex]\( x = 5 + i \)[/tex]
2. Understand the Multiplicity:
Each of these zeros appears once, so they have a multiplicity of 1.
3. Write the Polynomial in Factor Form:
Knowing the zeros, we can express the polynomial in its completely factored form:
[tex]\[
f(x) = -3 \left( x + \frac{1}{3} \right) (x - 1) (x - (5 - i)) (x - (5 + i))
\][/tex]
4. Simplify the Factored Form:
To simplify, start by adjusting the factor [tex]\( x + \frac{1}{3} \)[/tex] to avoid fractions:
[tex]\[
-3\left(x + \frac{1}{3}\right) = (-3)(x + \frac{1}{3}) = -3x - 1
\][/tex]
The final completely factored form is:
[tex]\[
f(x) = (-3x - 1)(x - 1)(x - (5 - i))(x - (5 + i))
\][/tex]
This gives us the polynomial expressed in factored form showcasing all its complex zeros.
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