(x^3+3x^2-4x-6)/(x^2-4) Long Division

Jasonbradley

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Jun 17, 2024

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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

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

2. 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.

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

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

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

6. Repeat steps 2-5: 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.

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

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

9. 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.