Divide using synthetic division:  `(x^4-3x^3-15x^2+19x+30)` `-:`  `(x-5)`

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`(x^4 - 3x^3 - 15x^2 + 19x + 30 ) -: (x-5)`

Using synthetic division, we would have the coefficients of x^4-3x^3-15x^2+19x+30 as our dividend and the root of x-5 as the divisor.

  `5`  `|`  `1`   `-3`   `-15`   `19`   `30`                                

      `---------------`

         `1`                                       Bring down 1. Then multiply 1 and 5.

                                                 Write the product at the bottom of -3.

 

  `5`  `|`  `1`   `-3`   `-15`   `19`   `30`                                

                `5`                               Add -3 and 5.  

      `---------------`   Then, multiply the sum with 5.

          `1 `    `2 `                               Write the product below -15.

 

  `5`  `|`  `1`   `-3`   `-15`   `19`   `30`                                

                 `5`      `10`                     Add -15 and 10.  

      `---------------`   Then, multiply the sum with 5.

         `1`      `2`      `5`                      Write the product below 19.

 

  `5`  `|`  `1`   `-3`   `-15`    `19`    `30`                                

                 `5`      `10`  `-25`             Add 19 and 25.  

      `---------------`   Then, multiply the sum with 5.

         `1`      `2`      -`5`    `-6`            Write the product below 30.

 

  `5`  `|`  `1`   `-3`  ` -15`    `19`      `30`                                

                 `5`     `10`  `-25`   `-30 `    Add -30 and -30.  

      `----------------`               

          `1`    `2`   `-5`     `-6`       `0`     These are the coefficients of the

                                                 quotient, with a degree of 3.

Hence, `(x^4-3x^3-15x^2+19x+30) -: (x-5) = x^3+2x^2-5x-6` .

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