(4x4+18x3+20x2+10x+8)÷(x+2)

5 min read Jun 16, 2024
(4x4+18x3+20x2+10x+8)÷(x+2)

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)

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