(2x^3+7x^2-6x-8)/(x+4) Quizizz

Jasonbradley

5 min read

·

Jun 16, 2024

30

Share

Dividing Polynomials: (2x^3+7x^2-6x-8)/(x+4)

This article will guide you through the process of dividing the polynomial (2x^3+7x^2-6x-8) by the binomial (x+4). This is a common topic in algebra, and understanding it is crucial for further mathematical exploration.

Understanding Polynomial Division

Polynomial division is similar to long division with numbers. It helps us determine how many times one polynomial (the divisor) goes into another polynomial (the dividend) and gives us a quotient and a remainder.

Here's how to perform the division:

1. Set up the division:

      ___________
   x+4 | 2x^3 + 7x^2 - 6x - 8 
   

2. Focus on the leading terms:

   • Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x): 2x^3 / x = 2x^2
   • Write the result (2x^2) above the dividend, aligning it with the x^2 term.

      2x^2 ______
   x+4 | 2x^3 + 7x^2 - 6x - 8 
   

3. Multiply the divisor by the term you just wrote:

   • (x+4) * (2x^2) = 2x^3 + 8x^2
   • Write this result below the dividend, aligning like terms.

      2x^2 ______
   x+4 | 2x^3 + 7x^2 - 6x - 8 
          2x^3 + 8x^2 
   

4. Subtract:

   • Subtract the expression you just wrote from the dividend. Remember to change the signs of the second expression.

      2x^2 ______
   x+4 | 2x^3 + 7x^2 - 6x - 8 
          2x^3 + 8x^2 
          -------
               -x^2 - 6x 
   

5. Bring down the next term:

   • Bring down the next term (-6x) from the dividend.

      2x^2 ______
   x+4 | 2x^3 + 7x^2 - 6x - 8 
          2x^3 + 8x^2 
          -------
               -x^2 - 6x - 8 
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (-x^2) by the leading term of the divisor (x): -x^2 / x = -x
   • Write -x above the dividend.
   • Multiply the divisor (x+4) by -x: -x^2 - 4x
   • Write the result below the new dividend.
   • Subtract, changing the signs.
   • Bring down the next term (-8).

      2x^2 -x ____
   x+4 | 2x^3 + 7x^2 - 6x - 8 
          2x^3 + 8x^2 
          -------
               -x^2 - 6x - 8 
               -x^2 - 4x 
               -------
                     -2x - 8
   

7. Final step:

   • Divide the leading term of the new dividend (-2x) by the leading term of the divisor (x): -2x / x = -2
   • Write -2 above the dividend.
   • Multiply the divisor (x+4) by -2: -2x - 8
   • Write the result below the new dividend.
   • Subtract.

      2x^2 -x -2 _
   x+4 | 2x^3 + 7x^2 - 6x - 8 
          2x^3 + 8x^2 
          -------
               -x^2 - 6x - 8 
               -x^2 - 4x 
               -------
                     -2x - 8
                     -2x - 8 
                     -------
                           0
   

8. Result:

   • The quotient is 2x^2 - x - 2 and the remainder is 0.

Conclusion

We successfully divided the polynomial (2x^3+7x^2-6x-8) by (x+4). The result shows that (x+4) is a factor of (2x^3+7x^2-6x-8) because the remainder is zero.