Solve each polynomial equation in the complex numbers.
[tex]2 x^4-7 x^3+38 x^2-47 x-34=0[/tex]
Find the complex zeros of [tex]f[/tex].
[tex]x=[/tex] ____
(Type an exact answer, using radicals as needed. Express complex numbers in terms of [tex]i[/tex]. Use a comma to separate answers as needed.)
Answer (2)
Use the Rational Root Theorem to identify potential rational roots.
Test the potential roots to find two rational roots: − 2 1 and 2 .
Perform polynomial division to reduce the quartic polynomial to a quadratic polynomial: 2 x 2 + 34 = 0 .
Solve the quadratic equation to find the remaining complex roots: ± i 17 .
The complex zeros are − 2 1 , 2 , i 17 , − i 17
Explanation
Problem Analysis We are given the polynomial equation 2 x 4 − 7 x 3 + 38 x 2 − 47 x − 34 = 0 and asked to find all complex zeros.
Rational Root Theorem We will use the Rational Root Theorem to find possible rational roots. The possible rational roots are ± 1 , ± 2 , ± 17 , ± 34 , ± 2 1 , ± 2 17 .
Finding Rational Roots By testing these possible roots, we find that x = − 2 1 and x = 2 are roots of the polynomial.
Polynomial Division Now we perform polynomial division to divide the polynomial by ( x + 2 1 ) ( x − 2 ) = x 2 − 2 3 x − 1 . Multiplying by 2 to clear the fraction, we divide by 2 x 2 − 3 x − 2 . After performing polynomial long division, we obtain 2 x 4 − 7 x 3 + 38 x 2 − 47 x − 34 = ( 2 x 2 − 3 x − 2 ) ( x 2 − 2 x + 17 ) . However, it's easier to perform synthetic division twice. First, divide by ( x + 2 1 ) and then divide the result by ( x − 2 ) .
Dividing 2 x 4 − 7 x 3 + 38 x 2 − 47 x − 34 by ( x + 2 1 ) gives 2 x 3 − 8 x 2 + 42 x − 68 . Dividing 2 x 3 − 8 x 2 + 42 x − 68 by ( x − 2 ) gives 2 x 2 + 0 x + 34 = 2 x 2 + 34 .
Solving the Quadratic So we have ( x + 2 1 ) ( x − 2 ) ( 2 x 2 + 34 ) = 0 . Thus, 2 x 2 + 34 = 0 , which gives x 2 = − 17 , so x = ± i 17 .
Final Answer Therefore, the complex zeros of the polynomial are x = − 2 1 , 2 , i 17 , − i 17 .
Stating the Solution The complex zeros of f are x = − 2 1 , 2 , i 17 , − i 17 .
Examples
Polynomial equations are used in various fields such as physics, engineering, and economics to model complex systems. For example, in physics, polynomial equations can describe the trajectory of a projectile or the behavior of electrical circuits. In economics, they can be used to model cost functions or revenue functions. Solving these equations helps in understanding and predicting the behavior of these systems, allowing for better design and decision-making. The ability to find both real and complex roots is crucial for a complete analysis of these models.
The complex zeros of the polynomial 2 x 4 − 7 x 3 + 38 x 2 − 47 x − 34 = 0 are x = − 2 1 , 2 , i 17 , − i 17 . This includes two rational roots and two complex roots derived from solving the reduced polynomial. Combining all found roots gives a complete set of solutions for the original quartic equation.
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