(2x^3-3x^2-18x-8) Divided By (x-4)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2x^3 - 3x^2 - 18x - 8) ÷ (x - 4)

This article will guide you through the process of dividing the polynomial (2x^3 - 3x^2 - 18x - 8) by (x - 4) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, much like long division is used for dividing numbers. The process involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend. This process is repeated until the degree of the remaining polynomial is less than the degree of the divisor.

Step-by-Step Solution

Let's perform the division:

1. Set up the problem:

             ____________
x - 4 | 2x^3 - 3x^2 - 18x - 8 

2. Divide the leading terms:

• Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x): 2x^3 / x = 2x^2.
• Write this result above the dividend.

             2x^2       
x - 4 | 2x^3 - 3x^2 - 18x - 8 

3. Multiply the divisor by the result:

• Multiply (x - 4) by 2x^2: (x - 4) * 2x^2 = 2x^3 - 8x^2.
• Write the result below the dividend.

             2x^2       
x - 4 | 2x^3 - 3x^2 - 18x - 8 
        2x^3 - 8x^2     

4. Subtract:

• Subtract the product from the dividend: (2x^3 - 3x^2) - (2x^3 - 8x^2) = 5x^2.
• Bring down the next term (-18x).

             2x^2       
x - 4 | 2x^3 - 3x^2 - 18x - 8 
        2x^3 - 8x^2     
        ------------
             5x^2 - 18x

5. Repeat steps 2-4:

• Divide the leading term of the new dividend (5x^2) by the leading term of the divisor (x): 5x^2 / x = 5x.
• Multiply (x - 4) by 5x: (x - 4) * 5x = 5x^2 - 20x.
• Subtract the product: (5x^2 - 18x) - (5x^2 - 20x) = 2x.
• Bring down the next term (-8).

             2x^2 + 5x     
x - 4 | 2x^3 - 3x^2 - 18x - 8 
        2x^3 - 8x^2     
        ------------
             5x^2 - 18x
             5x^2 - 20x 
             ------------
                     2x - 8

6. Repeat steps 2-4 again:

• Divide the leading term of the new dividend (2x) by the leading term of the divisor (x): 2x / x = 2.
• Multiply (x - 4) by 2: (x - 4) * 2 = 2x - 8.
• Subtract the product: (2x - 8) - (2x - 8) = 0.

             2x^2 + 5x + 2    
x - 4 | 2x^3 - 3x^2 - 18x - 8 
        2x^3 - 8x^2     
        ------------
             5x^2 - 18x
             5x^2 - 20x 
             ------------
                     2x - 8
                     2x - 8
                     ------------
                          0

Conclusion

Therefore, (2x^3 - 3x^2 - 18x - 8) ÷ (x - 4) = 2x^2 + 5x + 2. This means that the quotient is 2x^2 + 5x + 2, and the remainder is 0.