(iv) (3a^(5)-2a^(4)+a^(3)-2) By (a^(2)+a+1)

Jasonbradley

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Jun 16, 2024

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

Polynomial long division is a method used to divide polynomials, similar to long division with numbers. Here's how to divide (3a^(5)-2a^(4)+a^(3)-2) by (a^(2)+a+1):

Steps:

1. Set up the division:

        ____________
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
   

2. Divide the leading terms:

   • The leading term of the divisor (a^(2)) goes into the leading term of the dividend (3a^(5)) 3a^(3) times.
   • Write 3a^(3) above the dividend.

        3a^3 _______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
   

3. Multiply the divisor by the quotient term:

   • Multiply (a^(2)+a+1) by 3a^(3) to get 3a^(5) + 3a^(4) + 3a^(3).

        3a^3 _______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
            3a^5 + 3a^4 + 3a^3
   

4. Subtract:

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

        3a^3 _______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
            3a^5 + 3a^4 + 3a^3
            -----------------
                 -5a^4 - 2a^3 
   

5. Bring down the next term:

   • Bring down the next term of the dividend (-2).

        3a^3 _______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
            3a^5 + 3a^4 + 3a^3
            -----------------
                 -5a^4 - 2a^3 - 2 
   

6. Repeat steps 2-5:

   • The leading term of the divisor (a^(2)) goes into the leading term of the new dividend (-5a^(4)) -5a^(2) times.
   • Multiply the divisor (a^(2)+a+1) by -5a^(2) and subtract the result.
   • Bring down the next term (-2).

        3a^3 - 5a^2 ______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
            3a^5 + 3a^4 + 3a^3
            -----------------
                 -5a^4 - 2a^3 - 2 
                 -5a^4 - 5a^3 - 5a^2
                 --------------------
                       3a^3 + 5a^2 - 2
   

7. Repeat steps 2-5 again:

   • The leading term of the divisor (a^(2)) goes into the leading term of the new dividend (3a^(3)) 3a times.
   • Multiply the divisor (a^(2)+a+1) by 3a and subtract the result.

        3a^3 - 5a^2 + 3a _______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
            3a^5 + 3a^4 + 3a^3
            -----------------
                 -5a^4 - 2a^3 - 2 
                 -5a^4 - 5a^3 - 5a^2
                 --------------------
                       3a^3 + 5a^2 - 2
                       3a^3 + 3a^2 + 3a
                       ------------------
                             2a^2 - 3a - 2
   

8. Repeat steps 2-5 one last time:

   • The leading term of the divisor (a^(2)) goes into the leading term of the new dividend (2a^(2)) 2 times.
   • Multiply the divisor (a^(2)+a+1) by 2 and subtract the result.

        3a^3 - 5a^2 + 3a + 2 _______
   a^2+a+1 | 3a^5 - 2a^4 + a^3  - 2 
            3a^5 + 3a^4 + 3a^3
            -----------------
                 -5a^4 - 2a^3 - 2 
                 -5a^4 - 5a^3 - 5a^2
                 --------------------
                       3a^3 + 5a^2 - 2
                       3a^3 + 3a^2 + 3a
                       ------------------
                             2a^2 - 3a - 2
                             2a^2 + 2a + 2
                             ------------
                                  -5a - 4
   

9. The remainder:

   • The final result is -5a-4. This is the remainder, and it has a degree less than the divisor.

Therefore, the result of dividing (3a^(5)-2a^(4)+a^(3)-2) by (a^(2)+a+1) is:

3a^(3) - 5a^(2) + 3a + 2 with a remainder of -5a-4

This can be written as:

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