(2x3−3x2+6x+4)÷(2x+1)

3 min read Jun 16, 2024
(2x3−3x2+6x+4)÷(2x+1)

Polynomial Division: (2x³−3x²+6x+4) ÷ (2x+1)

This article will guide you through the process of dividing the polynomial (2x³−3x²+6x+4) by the binomial (2x+1).

Long Division Method

  1. Set up the problem: Write the polynomials in a long division format.

         ____________
    2x+1 | 2x³ - 3x² + 6x + 4
    
  2. Divide the leading terms: Divide the leading term of the dividend (2x³) by the leading term of the divisor (2x). This gives us .

         x² _________
    2x+1 | 2x³ - 3x² + 6x + 4
    
  3. Multiply and subtract: Multiply the divisor (2x+1) by the quotient term (x²) and subtract the result from the dividend.

         x² _________
    2x+1 | 2x³ - 3x² + 6x + 4 
            -(2x³ + x²)
            ----------------
                 -4x² + 6x 
    
  4. Bring down the next term: Bring down the next term from the dividend (6x).

         x² _________
    2x+1 | 2x³ - 3x² + 6x + 4 
            -(2x³ + x²)
            ----------------
                 -4x² + 6x 
                 -4x² + 6x
    
  5. Repeat steps 2-4: Divide the new leading term (-4x²) by the leading term of the divisor (2x). This gives us -2x. Multiply the divisor by this quotient term and subtract. Bring down the next term (4).

         x² - 2x _______
    2x+1 | 2x³ - 3x² + 6x + 4 
            -(2x³ + x²)
            ----------------
                 -4x² + 6x 
                 -4x² + 6x
                 --------------
                       +4 
    
  6. Final step: Divide the new leading term (4) by the leading term of the divisor (2x). This gives us 2. Multiply the divisor by this quotient term and subtract.

         x² - 2x + 2 _______
    2x+1 | 2x³ - 3x² + 6x + 4 
            -(2x³ + x²)
            ----------------
                 -4x² + 6x 
                 -4x² + 6x
                 --------------
                       +4 
                       -(4 + 2)
                       -------
                           2
    

Therefore, the quotient is (x² - 2x + 2) and the remainder is 2.

The complete division can be expressed as:

(2x³−3x²+6x+4) ÷ (2x+1) = (x² - 2x + 2) + 2/(2x+1)

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