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

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: Finding the Quotient and Remainder

This article will guide you through the process of dividing the polynomial (2x⁴ + 5x³ + 3x² + 8x + 12) by the polynomial (2x + 3), determining the quotient and remainder.

Polynomial Long Division

The process of dividing polynomials resembles the long division of numbers. Let's break down the steps:

1. Set up the division:

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

2. Divide the leading terms:

   • Divide the leading term of the dividend (2x⁴) by the leading term of the divisor (2x). This gives us x³.
   • Write x³ above the dividend.

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

3. Multiply the divisor by the term just written:

   • Multiply (2x + 3) by x³ to get 2x⁴ + 3x³.
   • Write this result below the dividend.

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

4. Subtract:

   • Subtract the result from the dividend.

        x³ ___________
   2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
           2x⁴ + 3x³
           ---------
                 2x³ + 3x² 
   

5. Bring down the next term:

   • Bring down the next term from the dividend (3x²).

        x³ ___________
   2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
           2x⁴ + 3x³
           ---------
                 2x³ + 3x² + 8x 
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (2x³) by the leading term of the divisor (2x). This gives us x².
   • Write x² above the dividend.
   • Multiply (2x + 3) by x² to get 2x³ + 3x².
   • Subtract the result from the new dividend.
   • Bring down the next term (8x).

        x³ + x² ______
   2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
           2x⁴ + 3x³
           ---------
                 2x³ + 3x² + 8x 
                 2x³ + 3x²
                 ---------
                       5x + 12 
   

7. Continue the process:

   • Divide the leading term of the new dividend (5x) by the leading term of the divisor (2x). This gives us 5/2.
   • Write 5/2 above the dividend.
   • Multiply (2x + 3) by 5/2 to get 5x + 15/2.
   • Subtract the result from the new dividend.

        x³ + x² + 5/2 ____
   2x + 3 | 2x⁴ + 5x³ + 3x² + 8x + 12 
           2x⁴ + 3x³
           ---------
                 2x³ + 3x² + 8x 
                 2x³ + 3x²
                 ---------
                       5x + 12 
                       5x + 15/2
                       -------
                         9/2
   

8. The quotient and remainder:

   • The quotient is the polynomial above the division line: x³ + x² + 5/2.
   • The remainder is the term below the line: 9/2.

Verification

We can verify our solution by multiplying the quotient by the divisor and adding the remainder:

(x³ + x² + 5/2) * (2x + 3) + 9/2 = 2x⁴ + 5x³ + 3x² + 8x + 12

This confirms that our quotient and remainder are correct.

Therefore, the division of (2x⁴ + 5x³ + 3x² + 8x + 12) by (2x + 3) yields:

• Quotient: x³ + x² + 5/2
• Remainder: 9/2