Verified Answer :

  • Identify the dividend \(P(x) = x^3 + 3x^2 + 48\) and the divisor \(D(x) = x+5\).
  • Apply polynomial long division or synthetic division using the root \(x = -5\).
  • Determine the quotient to be \(Q(x) = x^2 - 2x + 10\) and the remainder to be \(R = -2\).
  • The final result is: Quotient: \(\boxed{x^2 - 2x + 10}\), Remainder: \(\boxed{-2}\).

Explanation

  1. Analysis of the Polynomial Division Problem

The problem requires dividing the polynomial \(P(x) = x^3 + 3x^2 + 48\) by the linear binomial \(D(x) = x + 5\). The goal is to express the fraction in the form \(Q(x) + \frac{R}{D(x)}\), where \(Q(x)\) is the quotient and \(R\) is the remainder. In this case, \(P(x)\) has coefficients \(1, 3, 0, 48\) (noting the placeholder for the \(x^1\) term), and the divisor is \(x + 5\). We will use synthetic division or polynomial long division to find these components.

  1. Performing the Division

Using synthetic division, we set the root of the divisor \(x+5=0\) to \(c = -5\).

  1. The coefficients of \(P(x)\) are \(1, 3, 0, 48\).
  2. Bring down the first coefficient: \(1\).
  3. Multiply by \(-5\) and add to the second coefficient: \(-5 \times 1 + 3 = -2\).
  4. Multiply \(-2\) by \(-5\) and add to the third coefficient: \(-5 \times (-2) + 0 = 10\).
  5. Multiply \(10\) by \(-5\) and add to the fourth coefficient: \(-5 \times 10 + 48 = -2\).

The resulting coefficients for the quotient are \(1, -2, 10\), which correspond to the quadratic \(Q(x) = x^2 - 2x + 10\), and the remainder is \(R = -2\).

  1. Conclusion and Final Result

The division of the polynomials leads to the result:
\[\frac{x^3 + 3x^2 + 48}{x + 5} = x^2 - 2x + 10 - \frac{2}{x + 5}\]
The quotient is \(x^2 - 2x + 10\) and the remainder is \(-2\).

Examples

Polynomial division is a fundamental tool in algebra used for simplifying complex expressions and solving equations. In physics, when analyzing the motion of an object under specific forces (like drag or variable thrust), we often derive equations that involve ratios of polynomials. By performing long division, engineers can reduce these ratios into simpler forms that are easier to integrate or graph, allowing them to predict the velocity or position of the object accurately.

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