Verified Answer :

  • The given coefficients \(2, -5, 1, 0\) represent the polynomial \(P(x) = 2x^3 - 5x^2 + x\).
  • Synthetic division requires a divisor \(c\) (the value \(x-c\)) which is missing from the provided table.
  • Using the Remainder Theorem, the remainder is equal to \(P(c)\).
  • Therefore, the remainder is given by the expression \(\boxed{2c^3 - 5c^2 + c}\).

Explanation

  1. Analysis of the Synthetic Division Table

Synthetic division is a shorthand method of polynomial division, typically used when dividing a polynomial \(P(x)\) by a linear factor of the form \(x - c\). The coefficients of the polynomial \(P(x) = 2x^3 - 5x^2 + x + 0\) are listed in the top row of the table. To complete the division, we need the divisor value, denoted as \(c\), which is usually placed to the left of the coefficients in the synthetic division table. As this value is not provided, the remainder depends on the chosen divisor \(c\).

  1. Calculating the Remainder as a Function of the Divisor

To perform the synthetic division for a general divisor \(c\), we follow the standard algorithm:

  1. Bring down the first coefficient, \(2\).
  2. Multiply by \(c\) to get \(2c\). Add this to the next coefficient \(-5\) to get \(-5 + 2c\).
  3. Multiply this result by \(c\) to get \(c(-5 + 2c) = 2c^2 - 5c\). Add this to the next coefficient \(1\) to get \(2c^2 - 5c + 1\).
  4. Multiply this result by \(c\) to get \(c(2c^2 - 5c + 1) = 2c^3 - 5c^2 + c\). Add this to the final coefficient \(0\) to get \(2c^3 - 5c^2 + c\).

The final value obtained, \(2c^3 - 5c^2 + c\), is the remainder. This is consistent with the Remainder Theorem, which states that the remainder of dividing a polynomial \(P(x)\) by \((x - c)\) is equal to \(P(c)\). For our polynomial \(P(x) = 2x^3 - 5x^2 + x\), the remainder is indeed \(P(c) = 2c^3 - 5c^2 + c\).

  1. Conclusion on the Remainder

Since the value of the divisor \(c\) is not specified in the table, the remainder remains a function of \(c\). If a specific divisor were provided, we could calculate a numerical value for the remainder. Without it, the remainder is represented by the expression \(2c^3 - 5c^2 + c\), where \(c\) is the value used as the divisor. Thus, the remainder cannot be determined as a single constant value without the divisor \(c\).

Examples

In computer science and algebra, synthetic division is often used to efficiently evaluate polynomials at specific points or to factorize expressions. For example, if an engineer is modeling a physical system with the polynomial \(P(c) = 2c^3 - 5c^2 + c\), they might use synthetic division to quickly find the value of the model at a specific point \(c\) without having to perform full long division or complex exponentiation manually. This saves computation time and minimizes errors in complex calculations.

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