(12x^3+2+11x+20x^2)/(2x+1)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (12x^3 + 2 + 11x + 20x^2) / (2x + 1)

This article will walk through the process of dividing the polynomial (12x^3 + 2 + 11x + 20x^2) by the binomial (2x + 1).

Understanding Polynomial Division

Polynomial division is similar to long division with numbers. We aim to find the quotient and remainder when dividing one polynomial by another.

Steps for Polynomial Division:

1. Arrange the polynomials: First, we need to arrange both the dividend (12x^3 + 2 + 11x + 20x^2) and the divisor (2x + 1) in descending order of their exponents. This gives us:

   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
   

2. Divide the leading terms: Divide the leading term of the dividend (12x^3) by the leading term of the divisor (2x). This gives us 6x^2.

       6x^2  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
   

3. Multiply the quotient by the divisor: Multiply the quotient (6x^2) by the divisor (2x + 1) to get 12x^3 + 6x^2.

       6x^2  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
           12x^3 + 6x^2
   

4. Subtract: Subtract the result (12x^3 + 6x^2) from the dividend.

       6x^2  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
           12x^3 + 6x^2
           -------------
                14x^2 + 11x 
   

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

       6x^2  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
           12x^3 + 6x^2
           -------------
                14x^2 + 11x 
   

6. Repeat steps 2-5: Now, repeat steps 2-5 with the new polynomial (14x^2 + 11x).

   • Divide the leading term (14x^2) by the leading term of the divisor (2x), which gives us 7x.
   • Multiply the quotient (7x) by the divisor (2x + 1), giving us 14x^2 + 7x.
   • Subtract this result from (14x^2 + 11x) to get 4x.

       6x^2 + 7x  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
           12x^3 + 6x^2
           -------------
                14x^2 + 11x 
                14x^2 + 7x 
                -------
                      4x + 2 
   

7. Bring down the last term: Bring down the last term of the dividend (2).

       6x^2 + 7x  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
           12x^3 + 6x^2
           -------------
                14x^2 + 11x 
                14x^2 + 7x 
                -------
                      4x + 2
   

8. Final Division: Repeat steps 2-5 one last time with the new polynomial (4x + 2).

   • Divide the leading term (4x) by the leading term of the divisor (2x), which gives us 2.
   • Multiply the quotient (2) by the divisor (2x + 1), giving us 4x + 2.
   • Subtract this result from (4x + 2), leaving a remainder of 0.

       6x^2 + 7x + 2  
   2x + 1 | 12x^3 + 20x^2 + 11x + 2 
           12x^3 + 6x^2
           -------------
                14x^2 + 11x 
                14x^2 + 7x 
                -------
                      4x + 2
                      4x + 2
                      -----
                        0 
   

Result

Therefore, the quotient of dividing (12x^3 + 2 + 11x + 20x^2) by (2x + 1) is 6x^2 + 7x + 2 and the remainder is 0.

This can be written as:

(12x^3 + 2 + 11x + 20x^2) / (2x + 1) = 6x^2 + 7x + 2