Verified Answer :

  • Perform polynomial long division by dividing the leading terms of the dividend and divisor at each step to determine the quotient terms.
  • Multiply the divisor by the new quotient terms and subtract from the current dividend.
  • Repeat the process until the degree of the remainder is strictly less than the degree of the divisor.
  • The final result is the quotient \(x + 3\) with a remainder of \(6\), which can be written as \(\boxed{x + 3 + \frac{6}{x^2 - 4}}\).

Explanation

  1. Problem Analysis

We are given the polynomial division problem to divide \(P(x) = x^3 + 3x^2 - 4x - 6\) by the quadratic divisor \(D(x) = x^2 - 4\). To perform the division, we look for a quotient \(Q(x)\) and a remainder \(R(x)\) such that \(P(x) = D(x)Q(x) + R(x)\), where the degree of \(R(x)\) is less than the degree of \(D(x)\). Here, \(D(x)\) is degree 2, so the remainder \(R(x)\) must be of degree 1 or less.

  1. Performing the First Division Step

To start, we divide the leading term of the dividend (\(x^3\)) by the leading term of the divisor (\(x^2\)):
\(\frac{x^3}{x^2} = x\).
This gives us the first term of our quotient. We then multiply the divisor by this term:
\(x \cdot (x^2 - 4) = x^3 - 4x\).
Next, we subtract this result from the original dividend:
\((x^3 + 3x^2 - 4x - 6) - (x^3 - 4x) = 3x^2 - 6\).
The result of this subtraction becomes our new current dividend.

  1. Completing the Division Process

Now we divide the leading term of our new dividend (\(3x^2\)) by the leading term of the divisor (\(x^2\)):
\(\frac{3x^2}{x^2} = 3\).
This gives us the second term of our quotient. We then multiply the divisor by this constant:
\(3 \cdot (x^2 - 4) = 3x^2 - 12\).
Finally, we subtract this from our current dividend:
\((3x^2 - 6) - (3x^2 - 12) = 6\).
Since the degree of the remainder \(6\) is 0, which is less than the degree of the divisor (degree 2), the division process is complete. The quotient is \(x+3\) and the remainder is \(6\).

  1. Final Result

Combining the results, the division of \(x^3 + 3x^2 - 4x - 6\) by \(x^2 - 4\) yields a quotient of \(x + 3\) and a remainder of \(6\). This can be expressed as:
\(\frac{x^3 + 3x^2 - 4x - 6}{x^2 - 4} = x + 3 + \frac{6}{x^2 - 4}\).
Thus, the final result is quotient \(x+3\) with remainder \(6\).

Examples

Polynomial division is a fundamental tool in algebra, used frequently in calculus for decomposing rational functions into simpler parts (partial fraction decomposition). For example, when integrating complex expressions like \(\int \frac{x^3 + 3x^2 - 4x - 6}{x^2 - 4} dx\), you would first use long division to simplify the integrand into \(\int (x + 3 + \frac{6}{x^2 - 4}) dx\), which is much easier to compute. This technique is essential for simplifying algebraic expressions in engineering, physics, and advanced mathematics.

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