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(3x^3-x^2-7x+6)/(x+2)

5 min read Jun 16, 2024
(3x^3-x^2-7x+6)/(x+2)

Dividing Polynomials: (3x^3-x^2-7x+6)/(x+2)

This article will guide you through the process of dividing the polynomial 3x^3-x^2-7x+6 by x+2. We will use polynomial long division to achieve this.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to how long division is used for numbers. Here's how it works for our given problem:

  1. Set up the division:

         __________
x+2 | 3x^3 - x^2 - 7x + 6

  2. Divide the leading terms:

    • Divide the leading term of the dividend (3x^3) by the leading term of the divisor (x). This gives us 3x^2.
    • Write 3x^2 above the line in the quotient.
         3x^2 _______
x+2 | 3x^3 - x^2 - 7x + 6

  3. Multiply the divisor by the quotient term:

    • Multiply (x+2) by 3x^2 to get 3x^3 + 6x^2.
         3x^2 _______
x+2 | 3x^3 - x^2 - 7x + 6 
        3x^3 + 6x^2

  4. Subtract:

    • Subtract the result from the dividend. Remember to change the signs of the terms in the second row before subtracting.
         3x^2 _______
x+2 | 3x^3 - x^2 - 7x + 6 
        3x^3 + 6x^2
       -----------
             -7x^2 - 7x

  5. Bring down the next term:

    • Bring down the next term from the dividend (-7x).
         3x^2 _______
x+2 | 3x^3 - x^2 - 7x + 6 
        3x^3 + 6x^2
       -----------
             -7x^2 - 7x

  6. Repeat steps 2-5:

    • Divide the leading term of the new dividend (-7x^2) by the leading term of the divisor (x), which gives us -7x.
    • Write -7x above the line in the quotient.
    • Multiply (x+2) by -7x to get -7x^2 - 14x.
    • Subtract this from the previous result.
    • Bring down the next term (+6).
         3x^2 - 7x _______
x+2 | 3x^3 - x^2 - 7x + 6 
        3x^3 + 6x^2
       -----------
             -7x^2 - 7x 
             -7x^2 - 14x
             -----------
                    7x + 6

  7. Final step:

    • Divide the leading term of the new dividend (7x) by the leading term of the divisor (x), which gives us 7.
    • Write 7 above the line in the quotient.
    • Multiply (x+2) by 7 to get 7x + 14.
    • Subtract this from the previous result.
         3x^2 - 7x + 7 _______
x+2 | 3x^3 - x^2 - 7x + 6 
        3x^3 + 6x^2
       -----------
             -7x^2 - 7x 
             -7x^2 - 14x
             -----------
                    7x + 6
                    7x + 14
                    --------
                           -8

  8. Result:

    • The quotient is 3x^2 - 7x + 7.
    • The remainder is -8.

Therefore, we can write the result as:

(3x^3 - x^2 - 7x + 6)/(x+2) = 3x^2 - 7x + 7 - 8/(x+2)

Conclusion

This example demonstrates how to divide polynomials using long division. By following these steps, you can divide any polynomial by another polynomial. Remember that the remainder will always be a polynomial with a degree lower than the divisor.

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