Solve each polynomial equation in the complex numbers:
[tex]3 x^4-26 x^3+66 x^2-50 x-25=0[/tex]
Find the complex zeros of [tex]$f$[/tex].
Answer (2)
Find possible rational roots using the Rational Root Theorem.
Identify rational roots by testing and use synthetic division to factor the polynomial.
Solve the remaining quadratic factor using the quadratic formula.
List all complex zeros: − 3 1 , 5 , 2 + i , 2 − i
Explanation
Finding Possible Rational Roots We are given the polynomial equation 3 x 4 − 26 x 3 + 66 x 2 − 50 x − 25 = 0 and asked to find all complex zeros. First, we can try to find rational roots using the Rational Root Theorem. Possible rational roots are ± 1 , ± 5 , ± 25 , ± 3 1 , ± 3 5 , ± 3 25 .
Identifying Rational Roots By testing these possible roots, we find that x = − 3 1 and x = 5 are roots of the polynomial. We can perform synthetic division or polynomial long division to factor the polynomial.
Factoring the Polynomial Dividing 3 x 4 − 26 x 3 + 66 x 2 − 50 x − 25 by ( x + 3 1 ) gives 3 x 3 − 27 x 2 + 75 x − 75 . Then, dividing 3 x 3 − 27 x 2 + 75 x − 75 by ( x − 5 ) gives 3 x 2 − 12 x + 15 . Thus, we have 3 x 4 − 26 x 3 + 66 x 2 − 50 x − 25 = ( 3 x + 1 ) ( x − 5 ) ( x 2 − 4 x + 5 ) .
Solving the Quadratic Factor Now we need to find the roots of the quadratic factor x 2 − 4 x + 5 = 0 . We can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c . In this case, a = 1 , b = − 4 , and c = 5 . So, x = 2 ( 1 ) 4 ± ( − 4 ) 2 − 4 ( 1 ) ( 5 ) = 2 4 ± 16 − 20 = 2 4 ± − 4 = 2 4 ± 2 i = 2 ± i .
Listing All Zeros Therefore, the complex zeros of the polynomial are x = − 3 1 , 5 , 2 + i , 2 − i .
Final Answer The complex zeros of the polynomial equation 3 x 4 − 26 x 3 + 66 x 2 − 50 x − 25 = 0 are x = − 3 1 , 5 , 2 + i , 2 − i .
Examples
Polynomial equations are used in various fields such as physics, engineering, and economics. For example, in physics, polynomial equations can be used to model the trajectory of a projectile. In engineering, they can be used to design structures and circuits. In economics, they can be used to model supply and demand curves. Solving polynomial equations helps in understanding and predicting the behavior of these systems.
The complex zeros of the polynomial 3 x 4 − 26 x 3 + 66 x 2 − 50 x − 25 = 0 are x = − 3 1 , 5 , 2 + i , 2 − i .
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