(11x+20x^2+12x^3+2)/(3x+2)

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (11x + 20x² + 12x³ + 2) / (3x + 2)

This article explores the process of polynomial long division, specifically focusing on dividing the polynomial 11x + 20x² + 12x³ + 2 by 3x + 2.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to long division with numbers. The goal is to find the quotient and remainder when one polynomial is divided by another.

Steps Involved

1. Set up the division:

   • Arrange the terms of both polynomials in descending order of their exponents.
   • Write the dividend (11x + 20x² + 12x³ + 2) inside the division symbol and the divisor (3x + 2) outside.

2. Divide the leading terms:

   • Divide the leading term of the dividend (12x³) by the leading term of the divisor (3x). This gives us 4x².
   • Write 4x² above the dividend, aligning it with the x² term.

3. Multiply the divisor by the quotient term:

   • Multiply the divisor (3x + 2) by the quotient term (4x²). This gives us 12x³ + 8x².
   • Write this result below the dividend, aligning terms with their corresponding exponents.

4. Subtract:

   • Subtract the result from step 3 from the dividend.
   • Remember to change the signs of the terms being subtracted.

5. Bring down the next term:

   • Bring down the next term of the dividend (11x).

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (12x²) by the leading term of the divisor (3x). This gives us 4x.
   • Write 4x above the dividend, aligning it with the x term.
   • Multiply the divisor by 4x: (3x + 2) * 4x = 12x² + 8x.
   • Subtract this result from the new dividend.
   • Bring down the next term (2).

7. Continue until the degree of the remainder is less than the degree of the divisor:

   • Repeat the process until the degree of the remainder is less than the degree of the divisor (3x + 2).

Solution

Following the steps outlined above, we arrive at the following solution:

             4x² + 4x - 2
     _______________________
3x + 2 | 12x³ + 20x² + 11x + 2 
         -(12x³ + 8x²)
         ----------------
                 12x² + 11x
                 -(12x² + 8x)
                 -------------
                         3x + 2
                         -(3x + 2)
                         -----------
                                  0 

Therefore, the quotient is 4x² + 4x - 2 and the remainder is 0.

Conclusion

We successfully divided the polynomial 11x + 20x² + 12x³ + 2 by 3x + 2 using polynomial long division. The result shows that the dividend is perfectly divisible by the divisor, resulting in a quotient of 4x² + 4x - 2 and a remainder of 0.