(2m3+m2+m+9)÷(2m-1)

Jasonbradley

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·

Jun 16, 2024

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Dividing Polynomials: (2m³ + m² + m + 9) ÷ (2m - 1)

This article explores the process of dividing the polynomial (2m³ + m² + m + 9) by (2m - 1) using long division.

Step-by-Step Breakdown:

1. Set up the long division:

        ___________
   2m-1 | 2m³ + m² + m + 9 
   

2. Divide the leading terms:

   • Divide the leading term of the dividend (2m³) by the leading term of the divisor (2m). This gives you m².
   • Write m² above the dividend.
   • Multiply the divisor (2m - 1) by m². This gives you 2m³ - m².
   • Subtract this result from the dividend:

        m²     
   2m-1 | 2m³ + m² + m + 9 
        -(2m³ - m²)
        ---------
              2m²  
   

3. Bring down the next term:

   • Bring down the next term from the dividend (+ m):

        m²     
   2m-1 | 2m³ + m² + m + 9 
        -(2m³ - m²)
        ---------
              2m² + m
   

4. Repeat the process:

   • Divide the new leading term (2m²) by the leading term of the divisor (2m). This gives you m.
   • Write m above the dividend.
   • Multiply the divisor (2m - 1) by m. This gives you 2m² - m.
   • Subtract this result from the previous result:

        m² + m 
   2m-1 | 2m³ + m² + m + 9 
        -(2m³ - m²)
        ---------
              2m² + m
              -(2m² - m)
              ---------
                     2m 
   

5. Bring down the next term:

   • Bring down the next term from the dividend (+ 9):

        m² + m 
   2m-1 | 2m³ + m² + m + 9 
        -(2m³ - m²)
        ---------
              2m² + m
              -(2m² - m)
              ---------
                     2m + 9
   

6. Repeat the process:

   • Divide the new leading term (2m) by the leading term of the divisor (2m). This gives you 1.
   • Write 1 above the dividend.
   • Multiply the divisor (2m - 1) by 1. This gives you 2m - 1.
   • Subtract this result from the previous result:

        m² + m + 1
   2m-1 | 2m³ + m² + m + 9 
        -(2m³ - m²)
        ---------
              2m² + m
              -(2m² - m)
              ---------
                     2m + 9
                     -(2m - 1)
                     ---------
                           10 
   

7. The remainder:

   • The final result is 10, which is the remainder.

Conclusion:

Therefore, the division of (2m³ + m² + m + 9) by (2m - 1) results in:

(2m³ + m² + m + 9) ÷ (2m - 1) = m² + m + 1 + 10/(2m - 1)