Use synthetic division to decide whether the given number is a solution of the equation.

5x^4 - 5x^2 + 5; x = 25

A. Yes
B. No

Using synthetic division on the polynomial 5 x 4 − 5 x 2 + 5 with x = 25 , we found a remainder of 1950005, which indicates that x = 25 is not a solution. Therefore, the answer is B. No .
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Answered by Anonymous | 2025-07-04

Set up synthetic division with the coefficients of the polynomial: 5, 0, -5, 0, 5 and the value x = 25.
Perform synthetic division.
Determine the remainder.
Since the remainder is 1950005, which is not 0, x = 25 is not a solution. The answer is N o ​ .

Explanation

Understanding the problem We are given the polynomial 5 x 4 − 5 x 2 + 5 and we want to determine if x = 25 is a solution to the equation 5 x 4 − 5 x 2 + 5 = 0 using synthetic division.

Setting up synthetic division To use synthetic division, we set up the coefficients of the polynomial. The coefficients are 5, 0, -5, 0, and 5. We will perform synthetic division with x = 25 .

Performing synthetic division We perform synthetic division as follows:


First, bring down the leading coefficient (5). Multiply 25 by 5 to get 125. Add this to the next coefficient (0) to get 125. Multiply 25 by 125 to get 3125. Add this to the next coefficient (-5) to get 3120. Multiply 25 by 3120 to get 78000. Add this to the next coefficient (0) to get 78000. Multiply 25 by 78000 to get 1950000. Add this to the last coefficient (5) to get 1950005.
The synthetic division process is:
25 ​ 5 5 ​ 0 125 125 ​ − 5 3125 3120 ​ 0 78000 78000 ​ 5 1950000 1950005 ​ ​

Determining if x=25 is a solution The remainder is 1950005. Since the remainder is not 0, x = 25 is not a solution to the equation 5 x 4 − 5 x 2 + 5 = 0 .

Verification by direct substitution Alternatively, we can directly substitute x = 25 into the polynomial to check if it equals zero:


5 ( 25 ) 4 − 5 ( 25 ) 2 + 5 = 5 ( 390625 ) − 5 ( 625 ) + 5 = 1953125 − 3125 + 5 = 1950005
Since the result is 1950005, which is not equal to 0, x = 25 is not a solution.

Final Answer Therefore, x = 25 is not a solution of the equation 5 x 4 − 5 x 2 + 5 = 0 .

Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . It's often used in engineering to analyze the stability of systems. For example, control systems engineers use polynomials to model the behavior of a system, and synthetic division can help quickly determine if a particular value is a root of the characteristic equation, indicating potential instability. This helps in designing stable and reliable control systems.

Answered by GinnyAnswer | 2025-07-04