(3x^3-5x^2+10x-3)/(3x+1)

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (3x^3-5x^2+10x-3) / (3x+1)

This article will guide you through the process of dividing the polynomial (3x^3-5x^2+10x-3) by (3x+1) using long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but with polynomials instead of numbers. It involves dividing a polynomial (the dividend) by another polynomial (the divisor) to find the quotient and the remainder.

Steps for Long Division

1. Set up the division: Write the dividend (3x^3-5x^2+10x-3) inside the division symbol and the divisor (3x+1) outside.

        __________
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
   

2. Divide the leading terms: Divide the leading term of the dividend (3x^3) by the leading term of the divisor (3x). This gives you x^2, which becomes the first term of the quotient. Write x^2 above the dividend.

        x^2     
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
   

3. Multiply the quotient term by the divisor: Multiply the quotient term (x^2) by the divisor (3x+1), which gives you 3x^3 + x^2. Write this below the dividend.

        x^2     
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
   

4. Subtract: Subtract the expression you just wrote from the dividend. This gives you -6x^2 + 10x - 3.

        x^2     
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
           ---------
                -6x^2 + 10x - 3 
   

5. Bring down the next term: Bring down the next term from the dividend (10x).

        x^2     
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
           ---------
                -6x^2 + 10x - 3 
                -6x^2
   

6. Repeat steps 2-5: Divide the new leading term (-6x^2) by the leading term of the divisor (3x) to get -2x. Write -2x in the quotient.

        x^2 - 2x
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
           ---------
                -6x^2 + 10x - 3 
                -6x^2 - 2x
   

   Multiply -2x by the divisor (3x+1) and write the result below the dividend. Subtract to get 12x - 3.

        x^2 - 2x
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
           ---------
                -6x^2 + 10x - 3 
                -6x^2 - 2x
                ---------
                     12x - 3 
   

7. Continue until the degree of the remainder is less than the degree of the divisor: Bring down the -3. Divide 12x by 3x to get 4 and add that to the quotient.

        x^2 - 2x + 4
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
           ---------
                -6x^2 + 10x - 3 
                -6x^2 - 2x
                ---------
                     12x - 3
                     12x + 4
   

   Multiply 4 by the divisor and subtract to find the remainder of -7.

        x^2 - 2x + 4
   3x+1 | 3x^3 - 5x^2 + 10x - 3 
           3x^3 + x^2
           ---------
                -6x^2 + 10x - 3 
                -6x^2 - 2x
                ---------
                     12x - 3
                     12x + 4
                     ---------
                          -7 
   

Result

The final result is: (3x^3-5x^2+10x-3) / (3x+1) = x^2 - 2x + 4 - 7/(3x+1)

This means the quotient is x^2 - 2x + 4 and the remainder is -7.