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$(8x^3-3x+1) Div(4x^3+x^2-2x-3)=$

5 min read Jun 16, 2024
$(8x^3-3x+1) Div(4x^3+x^2-2x-3)=$

Polynomial Long Division: $(8x^3-3x+1) \div (4x^3+x^2-2x-3)$

Polynomial long division is a method for dividing polynomials, similar to long division with numbers. Here's how to solve the division $(8x^3-3x+1) \div (4x^3+x^2-2x-3)$:

1. Set up the Division:

Write the problem like a long division problem:

             ____________
4x^3+x^2-2x-3 | 8x^3     -3x + 1

2. Divide the Leading Terms:

Divide the leading term of the dividend ($8x^3$) by the leading term of the divisor ($4x^3$):

  • $8x^3 / 4x^3 = 2$

Write the quotient (2) above the line:

             2        
4x^3+x^2-2x-3 | 8x^3     -3x + 1

3. Multiply and Subtract:

Multiply the divisor by the quotient (2):

  • $2 * (4x^3+x^2-2x-3) = 8x^3 + 2x^2 - 4x - 6$

Subtract this product from the dividend:

             2        
4x^3+x^2-2x-3 | 8x^3     -3x + 1 
             -(8x^3 + 2x^2 - 4x - 6)
             ----------------------
                   -2x^2 + x + 7

4. Bring Down the Next Term:

Bring down the next term of the dividend (-3x):

             2        
4x^3+x^2-2x-3 | 8x^3     -3x + 1 
             -(8x^3 + 2x^2 - 4x - 6)
             ----------------------
                   -2x^2 + x + 7 -3x

5. Repeat Steps 2-4:

Divide the leading term of the new dividend (-2x^2) by the leading term of the divisor (4x^3):

  • -2x^2 / 4x^3 = -1/2x

Write the quotient (-1/2x) above the line:

             2 -1/2x     
4x^3+x^2-2x-3 | 8x^3     -3x + 1 
             -(8x^3 + 2x^2 - 4x - 6)
             ----------------------
                   -2x^2 + x + 7 -3x

Multiply and subtract:

  • -1/2x * (4x^3+x^2-2x-3) = -2x^2 - 1/2x + x + 3/2
             2 -1/2x     
4x^3+x^2-2x-3 | 8x^3     -3x + 1 
             -(8x^3 + 2x^2 - 4x - 6)
             ----------------------
                   -2x^2 + x + 7 -3x
                   -(-2x^2 - 1/2x + x + 3/2)
                   ------------------------
                             3/2x + 13/2

Bring down the next term (1):

             2 -1/2x     
4x^3+x^2-2x-3 | 8x^3     -3x + 1 
             -(8x^3 + 2x^2 - 4x - 6)
             ----------------------
                   -2x^2 + x + 7 -3x
                   -(-2x^2 - 1/2x + x + 3/2)
                   ------------------------
                             3/2x + 13/2 + 1

6. Continue Until Degree of Remainder is Less than Divisor

Continue dividing and subtracting until the degree of the remainder is less than the degree of the divisor. In this case, the degree of the remainder (3/2x + 13/2) is 1, which is less than the degree of the divisor (3).

7. Express the Result:

The final result is:

$(8x^3-3x+1) \div (4x^3+x^2-2x-3) = \boxed{2 - \frac{1}{2}x + \frac{3}{2}x + \frac{13}{2} \div (4x^3+x^2-2x-3)}$

This means the quotient is $2 - \frac{1}{2}x$ and the remainder is $\frac{3}{2}x + \frac{13}{2}$.

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