(4x^3-3x^2+5x+6)/(x+6)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (4x^3 - 3x^2 + 5x + 6) / (x + 6)

This article will walk through the process of dividing the polynomial 4x^3 - 3x^2 + 5x + 6 by the binomial x + 6. We will use the long division method to achieve this.

Long Division Method

1. Set up the division: Write the dividend (4x^3 - 3x^2 + 5x + 6) inside the division symbol and the divisor (x + 6) outside.

        _________
   x + 6 | 4x^3 - 3x^2 + 5x + 6
   

2. Divide the leading terms: Divide the leading term of the dividend (4x^3) by the leading term of the divisor (x). This gives us 4x^2. Write this term above the division symbol.

        4x^2 ______
   x + 6 | 4x^3 - 3x^2 + 5x + 6
   

3. Multiply the divisor: Multiply the quotient term (4x^2) by the entire divisor (x + 6). This results in 4x^3 + 24x^2.

        4x^2 ______
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
   

4. Subtract: Subtract the result from the dividend. Be careful with the signs!

        4x^2 ______
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
           -----------
                  -27x^2 + 5x 
   

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

        4x^2 ______
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
           -----------
                  -27x^2 + 5x + 6
   

6. Repeat steps 2-5: Now, divide the leading term of the new dividend (-27x^2) by the leading term of the divisor (x). This gives us -27x. Write this term above the division symbol.

        4x^2 - 27x ______
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
           -----------
                  -27x^2 + 5x + 6
                  -27x^2 - 162x
   

   Multiply (-27x) by the divisor (x + 6), subtract, and bring down the next term (6).

        4x^2 - 27x ______
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
           -----------
                  -27x^2 + 5x + 6
                  -27x^2 - 162x
                  -----------
                          167x + 6 
   

7. Continue repeating steps 2-5: Divide 167x by x to get 167.

        4x^2 - 27x + 167 _____
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
           -----------
                  -27x^2 + 5x + 6
                  -27x^2 - 162x
                  -----------
                          167x + 6 
                          167x + 1002
   

   Multiply 167 by (x + 6), subtract, and we are left with the remainder.

        4x^2 - 27x + 167 _____
   x + 6 | 4x^3 - 3x^2 + 5x + 6
           4x^3 + 24x^2
           -----------
                  -27x^2 + 5x + 6
                  -27x^2 - 162x
                  -----------
                          167x + 6 
                          167x + 1002
                          -----------
                                 -996
   

8. The result: We can now write the result of the division as:

   4x^3 - 3x^2 + 5x + 6 = (x + 6)(4x^2 - 27x + 167) - 996

   Or, in the form of quotient and remainder:

   (4x^3 - 3x^2 + 5x + 6) / (x + 6) = 4x^2 - 27x + 167 - 996/(x + 6)

Conclusion

Using the long division method, we successfully divided the polynomial 4x^3 - 3x^2 + 5x + 6 by the binomial x + 6. The result is a quotient of 4x^2 - 27x + 167 and a remainder of -996.