(4x4+18x3+20x2+10x+8)÷(x+2)

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (4x^4 + 18x^3 + 20x^2 + 10x + 8) ÷ (x + 2)

This article will guide you through the process of dividing the polynomial 4x^4 + 18x^3 + 20x^2 + 10x + 8 by the binomial x + 2 using long division.

Steps for Polynomial Long Division

1. Set up the division. Write the dividend (4x^4 + 18x^3 + 20x^2 + 10x + 8) inside the division symbol and the divisor (x + 2) outside.

       ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
   

2. Divide the leading terms. Divide the leading term of the dividend (4x^4) by the leading term of the divisor (x). The result is 4x^3. Write this above the division symbol.

       4x^3 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
   

3. Multiply the divisor by the quotient term. Multiply (x + 2) by 4x^3, which gives 4x^4 + 8x^3. Write this result below the dividend.

       4x^3 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
   

4. Subtract. Subtract the result from the dividend.

       4x^3 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 
   

5. Bring down the next term. Bring down the next term of the dividend (20x^2).

       4x^3 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 + 20x^2 
   

6. Repeat steps 2-5. Divide the new leading term (10x^3) by the leading term of the divisor (x) to get 10x^2. Write this above the division symbol.

       4x^3 + 10x^2 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 + 20x^2 
   

   Multiply (x + 2) by 10x^2 and subtract the result.

       4x^3 + 10x^2 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 + 20x^2 
                 10x^3 + 20x^2
                 -----------
                          0
   

7. Bring down the next term. Bring down the next term of the dividend (10x).

       4x^3 + 10x^2 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 + 20x^2 
                 10x^3 + 20x^2
                 -----------
                          0 + 10x
   

8. Repeat steps 2-5. Divide (10x) by (x) to get 10.

       4x^3 + 10x^2 + 10 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 + 20x^2 
                 10x^3 + 20x^2
                 -----------
                          0 + 10x 
   

   Multiply (x + 2) by 10 and subtract.

       4x^3 + 10x^2 + 10 ___________________
   x + 2 | 4x^4 + 18x^3 + 20x^2 + 10x + 8 
          4x^4 + 8x^3
          -----------
                 10x^3 + 20x^2 
                 10x^3 + 20x^2
                 -----------
                          0 + 10x + 8 
                          10x + 20
                          -----
                               -12
   

9. The remainder is -12. Since the degree of the remainder (-12) is less than the degree of the divisor (x+2), we stop here.

Solution:

The result of dividing (4x^4 + 18x^3 + 20x^2 + 10x + 8) by (x + 2) is:

4x^3 + 10x^2 + 10 - (12/(x + 2))

This can also be written as:

4x^3 + 10x^2 + 10 + (-12)/(x + 2)