(x^3-13x^2+40x+18) Divided By (x-7)

Jasonbradley

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Jun 17, 2024

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Dividing Polynomials: (x^3 - 13x^2 + 40x + 18) ÷ (x - 7)

This article will guide you through the process of dividing the polynomial (x^3 - 13x^2 + 40x + 18) by the binomial (x - 7) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but instead of dividing numbers, you are dividing polynomials.

The Steps

1. Set up the problem: Write the dividend (x^3 - 13x^2 + 40x + 18) inside the division symbol and the divisor (x - 7) outside.

       ____________
   x - 7 | x^3 - 13x^2 + 40x + 18
   

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

       x^2 _______
   x - 7 | x^3 - 13x^2 + 40x + 18 
   

3. Multiply the quotient term by the divisor: Multiply x^2 by (x - 7), which gives you x^3 - 7x^2. Write this result below the dividend.

       x^2 _______
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
   

4. Subtract: Subtract the result from the dividend.

       x^2 _______
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 
   

5. Bring down the next term: Bring down the next term from the dividend (+40x).

       x^2 _______
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 + 40x
   

6. Repeat steps 2-5: Divide the leading term of the new dividend (-6x^2) by the leading term of the divisor (x). This gives you -6x. Write -6x above the 40x in the quotient.

       x^2 - 6x ______
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 + 40x
               -6x^2 + 42x
   

   Multiply -6x by (x - 7) and subtract the result.

       x^2 - 6x ______
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 + 40x
               -6x^2 + 42x
               -------
                      -2x
   

   Bring down the next term (+18).

       x^2 - 6x ______
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 + 40x
               -6x^2 + 42x
               -------
                      -2x + 18
   

7. Repeat steps 2-5: Divide the leading term of the new dividend (-2x) by the leading term of the divisor (x). This gives you -2. Write -2 above the 18 in the quotient.

       x^2 - 6x - 2 _____
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 + 40x
               -6x^2 + 42x
               -------
                      -2x + 18
                      -2x + 14
   

   Multiply -2 by (x - 7) and subtract the result.

       x^2 - 6x - 2 _____
   x - 7 | x^3 - 13x^2 + 40x + 18 
          x^3 - 7x^2
          -------
               -6x^2 + 40x
               -6x^2 + 42x
               -------
                      -2x + 18
                      -2x + 14
                      -------
                           4
   

8. The Remainder: The final result is 4. Since it is less than the degree of the divisor, we call it the remainder.

The Result

Therefore, when you divide (x^3 - 13x^2 + 40x + 18) by (x - 7), you get:

(x^3 - 13x^2 + 40x + 18) ÷ (x - 7) = x^2 - 6x - 2 + 4/(x - 7)