Use synthetic division to decide whether the given number is a solution of the given equation.
[tex]5 x^4-5 x^2+5 ; x=\frac{2}{5}[/tex]
Answer (2)
Using synthetic division, we determined that the remainder when substituting x = 5 2 into the polynomial 5 x 4 − 5 x 2 + 5 is 125 541 . Since the remainder is not 0, 5 2 is not a solution to the equation. Therefore, the final answer is N o .
;
Set up synthetic division with the coefficients of the polynomial and the potential root.
Perform the synthetic division process.
Determine the remainder after synthetic division.
Since the remainder 125 541 is not 0, the number 5 2 is not a solution. N o
Explanation
Understanding the Problem We are given the polynomial 5 x 4 − 5 x 2 + 5 and asked to determine if x = 5 2 is a solution to the equation 5 x 4 − 5 x 2 + 5 = 0 using synthetic division.
Setting up Synthetic Division We will use synthetic division to test if 5 2 is a root of the polynomial. We set up the synthetic division with 5 2 as the divisor and the coefficients of the polynomial as the dividend. The coefficients are 5, 0, -5, 0, 5.
Performing Synthetic Division Performing synthetic division:
Bring down the first coefficient (5).
Multiply the root 5 2 by 5 to get 2.
Add 2 to the next coefficient (0) to get 2.
Multiply the root 5 2 by 2 to get 5 4 .
Add 5 4 to the next coefficient (-5) to get − 5 21 .
Multiply the root 5 2 by − 5 21 to get − 25 42 .
Add − 25 42 to the next coefficient (0) to get − 25 42 .
Multiply the root 5 2 by − 25 42 to get − 125 84 .
Add − 125 84 to the last coefficient (5) to get 5 − 125 84 = 125 625 − 84 = 125 541 .
Determining the Solution The remainder is 125 541 , which is approximately 4.328. Since the remainder is not 0, 5 2 is not a solution to the equation 5 x 4 − 5 x 2 + 5 = 0 .
Final Answer Therefore, 5 2 is not a solution of the given equation.
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − c . It's often used in engineering to simplify complex polynomial expressions that arise in system modeling. For instance, when analyzing the stability of a control system, engineers might use synthetic division to quickly check if a particular value is a root of the characteristic equation, which helps determine the system's response to disturbances. This method streamlines the process of finding roots and understanding system behavior, saving time and resources in design and analysis.
