How many and of which kind of roots does the equation
f(x) = 2x^4 - 9x^3 + 17x^2 - 16x + 6 have?
A. 4 real; 4 complex
B. 4 real
C. 4 complex
D. 2 real; 2 complex
Answer (2)
The polynomial equation has 2 real roots (1.5 and 1) and 2 complex roots (1+i and 1-i). Therefore, the correct choice is 2 real; 2 complex. The answer is: D.
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The problem asks for the number and type of roots of the equation f ( x ) = 2 x 4 − 9 x 3 + 17 x 2 − 16 x + 6 = 0 .
We find the roots using numerical methods.
The roots are approximately 1 + i , 1 − i , 1.5 , and 1 .
Therefore, there are 2 real and 2 complex roots: 2 real ; 2 complex .
Explanation
Problem Analysis We are given the equation f ( x ) = 2 x 4 − 9 x 3 + 17 x 2 − 16 x + 6 = 0 and asked to determine the number and type of its roots.
Solution Strategy To find the roots, we can attempt to factor the polynomial or use numerical methods. In this case, we will use a numerical method to approximate the roots.
Finding the Roots Using a numerical method, we find the roots to be approximately 1 + i , 1 − i , 1.5 , and 1 . This means there are two real roots ( 1.5 and 1 ) and two complex roots ( 1 + i and 1 − i ).
Conclusion Therefore, the equation has 2 real roots and 2 complex roots.
Examples
Understanding the nature of roots (real or complex) is crucial in many engineering applications, such as control systems. For example, in designing a stable feedback control system, the roots of the characteristic equation determine the system's stability. Real roots indicate stable or unstable behavior, while complex roots indicate oscillatory behavior. By analyzing the roots, engineers can adjust system parameters to achieve desired performance, ensuring the system remains stable and responds appropriately to inputs. This ensures that systems like cruise control in cars or temperature regulation in homes function reliably.
