(4a^6-5a^4+3a^2-a)/(2a+1)

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (4a^6-5a^4+3a^2-a)/(2a+1)

In this article, we'll walk through the process of dividing the polynomial 4a^6 - 5a^4 + 3a^2 - a by the binomial 2a + 1 using polynomial long division.

Setting up the Division

1. Arrange the polynomials: Write the dividend (4a^6 - 5a^4 + 3a^2 - a) and the divisor (2a + 1) in a long division format. Ensure that both polynomials are arranged in descending order of their exponents.

        _________________________
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
   

2. Fill in missing terms: If there are any missing terms in the dividend (terms with exponents that are not present), we need to add them with a coefficient of 0. In this case, there are no missing terms.

Performing the Division

1. Divide the leading terms: Divide the leading term of the dividend (4a^6) by the leading term of the divisor (2a). This gives us 2a^5.

        2a^5 ______________________
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
   

2. Multiply the quotient by the divisor: Multiply 2a^5 by (2a + 1), which gives us 4a^6 + 2a^5.

        2a^5 ______________________
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
          -(4a^6 + 2a^5)
   

3. Subtract: Subtract the result (4a^6 + 2a^5) from the dividend.

        2a^5 ______________________
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
          -(4a^6 + 2a^5)
          ------------------
                -2a^5 - 5a^4 
   

4. Bring down the next term: Bring down the next term of the dividend (-5a^4).

        2a^5 ______________________
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
          -(4a^6 + 2a^5)
          ------------------
                -2a^5 - 5a^4 + 3a^2
   

5. Repeat the process: Repeat steps 1-4 with the new dividend (-2a^5 - 5a^4 + 3a^2). Divide the leading term (-2a^5) by the leading term of the divisor (2a) to get -a^4. Multiply -a^4 by (2a + 1) to get -2a^5 - a^4. Subtract this from the current dividend.

        2a^5 - a^4 _________________
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
          -(4a^6 + 2a^5)
          ------------------
                -2a^5 - 5a^4 + 3a^2
                -(-2a^5 - a^4)
                -----------------
                       -4a^4 + 3a^2 
   

6. Continue repeating: Continue this process until the degree of the remaining dividend is less than the degree of the divisor.

        2a^5 - a^4 + 2a^3 - a^2 + 2a - 1 
   2a+1 | 4a^6 - 5a^4 + 3a^2 - a 
          -(4a^6 + 2a^5)
          ------------------
                -2a^5 - 5a^4 + 3a^2
                -(-2a^5 - a^4)
                -----------------
                       -4a^4 + 3a^2 
                       -(-4a^4 - 2a^3)
                       -----------------
                             2a^3 + 3a^2 - a 
                             -(2a^3 + a^2)
                             -----------------
                                   2a^2 - a
                                   -(2a^2 + a)
                                   -----------------
                                        -2a
                                        -(-2a - 1)
                                        -----------------
                                              1 
   

Result

The result of the division is: 2a^5 - a^4 + 2a^3 - a^2 + 2a - 1 with a remainder of 1.

Therefore, we can express the original expression as:

(4a^6 - 5a^4 + 3a^2 - a)/(2a + 1) = 2a^5 - a^4 + 2a^3 - a^2 + 2a - 1 + 1/(2a + 1)