(2x3−3x2+6x+4)÷(2x+1)

Jasonbradley

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Jun 16, 2024

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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.

        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)