Long Division of Polynomials: (x^3 + 3x^2  4x  6) / (x^2  4)
This article will guide you through the process of performing long division with polynomials, specifically focusing on the division of (x^3 + 3x^2  4x  6) by (x^2  4).
Understanding Long Division of Polynomials
Long division with polynomials is similar to the long division you learned for numbers. The process involves dividing a polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder.
Steps for Long Division

Set up the division: Write the dividend (x^3 + 3x^2  4x  6) inside the division symbol and the divisor (x^2  4) outside.

Focus on the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x^2). This gives you the first term of the quotient, which is x.

Multiply: Multiply the quotient term (x) by the entire divisor (x^2  4), resulting in x(x^2  4) = x^3  4x.

Subtract: Subtract the result from the dividend. This gives you: (x^3 + 3x^2  4x  6)  (x^3  4x) = 3x^2  6

Bring down the next term: Bring down the next term from the dividend (6), resulting in 3x^2  6.

Repeat steps 25: Now, divide the new leading term (3x^2) by the leading term of the divisor (x^2), which gives you 3. This becomes the next term of the quotient.

Multiply: Multiply the new quotient term (3) by the divisor (x^2  4), resulting in 3(x^2  4) = 3x^2  12.

Subtract: Subtract the result from the current expression. This gives you: (3x^2  6)  (3x^2  12) = 6

The remainder: Since the degree of the remainder (6) is less than the degree of the divisor (x^2  4), we stop the process.
The Final Result
Therefore, the long division of (x^3 + 3x^2  4x  6) by (x^2  4) yields:
 Quotient: x + 3
 Remainder: 6
This can be expressed as:
(x^3 + 3x^2  4x  6) / (x^2  4) = x + 3 + 6/(x^2  4)
Conclusion
By following the steps above, you can successfully perform long division with polynomials. This method is crucial for simplifying expressions, solving equations, and gaining a deeper understanding of polynomial relationships.