(2x4+5x3+3x2+8x+12)÷(2x+3)

Jasonbradley

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Jun 16, 2024

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Simplifying Algebraic Expressions: (2x⁴ + 5x³ + 3x² + 8x + 12) ÷ (2x + 3)

This article will guide you through the process of simplifying the algebraic expression (2x⁴ + 5x³ + 3x² + 8x + 12) ÷ (2x + 3). We will use polynomial long division to achieve this.

Understanding Polynomial Long Division

Polynomial long division is similar to regular long division but involves dividing polynomials. Here's how it works:

1. Set up the division: Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial dividing) outside.
2. Divide the leading terms: Focus on the leading terms of both the dividend and divisor. Divide the leading term of the dividend by the leading term of the divisor. Write the result above the division symbol.
3. Multiply and subtract: Multiply the divisor by the result you just wrote. Write this product below the dividend and subtract it from the dividend.
4. Bring down the next term: Bring down the next term from the dividend.
5. Repeat steps 2-4: Repeat the process of dividing, multiplying, subtracting, and bringing down until you get a remainder that is either zero or has a lower degree than the divisor.

Applying Polynomial Long Division to Our Problem

Let's apply these steps to the given expression:

(2x⁴ + 5x³ + 3x² + 8x + 12) ÷ (2x + 3)

1. Set up the division:

             ________________________
2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 

2. Divide the leading terms:

(2x⁴) / (2x) = x³

Write x³ above the division symbol.

             x³ ______________________
2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 

3. Multiply and subtract:

(2x + 3) * x³ = 2x⁴ + 3x³

             x³ ______________________
2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
            -(2x⁴ + 3x³)
            -----------
                  2x³ 

4. Bring down the next term:

             x³ ______________________
2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
            -(2x⁴ + 3x³)
            -----------
                  2x³ + 3x²

5. Repeat steps 2-4:

• Divide the leading terms: (2x³) / (2x) = x²
• Multiply and subtract: (2x + 3) * x² = 2x³ + 3x²
• Bring down the next term:

             x³ + x² __________________
2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
            -(2x⁴ + 3x³)
            -----------
                  2x³ + 3x² + 8x 
                  -(2x³ + 3x²)
                  -----------
                        8x + 12

• Divide the leading terms: (8x) / (2x) = 4
• Multiply and subtract: (2x + 3) * 4 = 8x + 12
• Bring down the next term:

             x³ + x² + 4 ___________
2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
            -(2x⁴ + 3x³)
            -----------
                  2x³ + 3x² + 8x 
                  -(2x³ + 3x²)
                  -----------
                        8x + 12 
                        -(8x + 12)
                        ---------
                          0

Result

The result of dividing (2x⁴ + 5x³ + 3x² + 8x + 12) by (2x + 3) is x³ + x² + 4. The remainder is zero, indicating that (2x + 3) is a factor of the original polynomial.

Therefore, we can conclude:

(2x⁴ + 5x³ + 3x² + 8x + 12) ÷ (2x + 3) = x³ + x² + 4