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(4x^3+2x^2+3x+5)/(x^2+3x+1)

4 min read Jun 16, 2024
(4x^3+2x^2+3x+5)/(x^2+3x+1)

Dividing Polynomials: A Step-by-Step Guide

This article explores the division of polynomials, specifically focusing on the expression:

(4x^3 + 2x^2 + 3x + 5) / (x^2 + 3x + 1)

We will utilize the long division method to find the quotient and remainder of this division.

Understanding Long Division of Polynomials

Long division of polynomials is similar to long division with numbers. The process involves these steps:

  1. Set up the division: Write the dividend (4x^3 + 2x^2 + 3x + 5) inside the division symbol and the divisor (x^2 + 3x + 1) outside.

  2. Divide the leading terms: Focus on the leading terms of both the dividend and the divisor (4x^3 and x^2). Divide them (4x^3 / x^2 = 4x). This result (4x) becomes the first term of the quotient.

  3. Multiply and subtract: Multiply the divisor (x^2 + 3x + 1) by the obtained quotient term (4x) and write the result beneath the dividend. Subtract this product from the dividend.

  4. Bring down the next term: Bring down the next term of the dividend (3x).

  5. Repeat steps 2-4: Now, focus on the leading term of the new dividend and the divisor. Divide them, multiply, and subtract. Bring down the next term. Continue this process until the degree of the new dividend is less than the degree of the divisor.

Performing the Long Division

Let's apply these steps to our given expression:

             4x - 10
     x^2 + 3x + 1 | 4x^3 + 2x^2 + 3x + 5 
                 -(4x^3 + 12x^2 + 4x)
                 ---------------------
                       -10x^2 - x + 5
                       -(-10x^2 - 30x - 10)
                       ---------------------
                                29x + 15

Interpretation:

  • Quotient: The quotient is 4x - 10.
  • Remainder: The remainder is 29x + 15.

Therefore, the expression can be represented as:

(4x^3 + 2x^2 + 3x + 5) / (x^2 + 3x + 1) = (4x - 10) + (29x + 15) / (x^2 + 3x + 1)

Conclusion

We successfully performed the long division of polynomials and obtained the quotient and remainder. This process provides a clear way to divide polynomials, enabling further analysis and manipulation of the expression. Remember, the final result represents the original expression in a simplified form.

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