Verified Answer :

  • Set up the synthetic division using the coefficients of \(P(x) = x^3 + 0x^2 + 3x + 8\) and the divisor root \(-1\).
  • Perform the synthetic division to find the coefficients of the quotient: \(1\), \(-1\), and \(4\).
  • State the resulting quotient as \(Q(x) = x^2 - x + 4\).
  • Identify the remainder as \(4\).
  • The quotient is \(\boxed{x^2 - x + 4}\) and the remainder is \(\boxed{4}\).

Explanation

  1. Problem Analysis and Strategy

We are tasked with dividing the polynomial \(P(x) = x^3 + 3x + 8\) by the linear divisor \(D(x) = x + 1\). According to the Division Algorithm for polynomials, there exist a unique quotient \(Q(x)\) and a remainder \(R\) such that \(P(x) = D(x) \cdot Q(x) + R\), where the degree of \(R\) is less than the degree of \(D(x)\). Since \(D(x)\) has degree 1, the remainder \(R\) must be a constant. We can solve this using polynomial long division or synthetic division.

  1. Performing the Division

Using synthetic division, we use the root of the divisor \(x+1=0\), which is \(x = -1\). The coefficients of the polynomial \(x^3 + 0x^2 + 3x + 8\) are \(1\), \(0\), \(3\), and \(8\).

The process is as follows:

  • Bring down the leading coefficient, \(1\).
  • Multiply by the root \(-1\): \(1 \times (-1) = -1\). Add to the next coefficient: \(0 + (-1) = -1\).
  • Multiply by the root \(-1\): \((-1) \times (-1) = 1\). Add to the next coefficient: \(3 + 1 = 4\).
  • Multiply by the root \(-1\): \(4 \times (-1) = -4\). Add to the next coefficient: \(8 + (-4) = 4\).

The resulting values represent the coefficients of the quotient and the final remainder. The quotient is \(1x^2 - 1x + 4\), and the remainder is \(4\).

  1. Conclusion and Verification

We have verified that dividing \(x^3 + 3x + 8\) by \(x+1\) yields a quotient of \(x^2 - x + 4\) and a constant remainder of \(4\). Checking this: \((x+1)(x^2 - x + 4) + 4 = (x^3 - x^2 + 4x + x^2 - x + 4) + 4 = x^3 + 3x + 8\). This confirms our result is correct.
The quotient is \(\boxed{x^2 - x + 4}\).
The remainder is \(\boxed{4}\).

Examples

Polynomial division is essential in fields like control theory and signal processing, where it is used to simplify transfer functions or analyze system stability. By decomposing a rational function into a quotient and a remainder, engineers can better understand how a system's output responds to specific inputs. For example, when analyzing the electrical behavior of a circuit, this technique helps isolate the steady-state component (the quotient) from the transient, decaying component (the remainder).

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