Hey guys, Steffi again. This time I'm going to show you how to divide polynomials by other polynomials in two ways. On Tuesday, we divided polynomials using the Long division method, and on Wednesday, we learned how to divide polynomials using the Synthetic Division.

Option 1: Long division

Ex. Divide x² – 9x – 10 by x + 1

Step 1: Set up the division; making sure that the variables in both the dividend and the divisor are arranged in descending powers.



Step 2:  Look at the leading x of the divisor and the dividend. Divide the leading x of the dividend (x^2) by the leading x of the divisor (x), then put the answer ( which is x) on top as part of the quotient.

  

Step 3: Multiply that x (newly found quotient) to the divisor using Distributive Law and put that under the dividend, and then subtract each term. 

(x^2 - x^2 = 0; -9x - (+1x) = -10x )

                        * When using Long Division, always SUBTRACT.

Step 4: Bring down the last term (-10) to form the new dividend.

Step 5: Repeat steps 2 and 3 until the remainder is a degree lower than the divisor.

      ex. x^3 is a degree lower than x^4; 10 (a constant) is a degree lower than x (a variable). 

        To check: Quotient * Divisor

                         (x -10) (x+1)

                      =  x^2 +1x -10x -10

                      =  x^2 -9x -10  [Dividend]

        Quotient:  x-10

            Roots:  x = 10 and x = -1   (from the quotient and the divisor)

Example with a remainder other than 0: 

     To check:  Quotient * Divisor + Remainder = Dividend

                      (x+2) (x+1) + 3

                    =  x^2 + 1x +2x +2 +3

                    =  x^2 + 3x +5    [Dividend]

    Quotient:  x + 2

        Roots:  x = -2 and x = -1

 Remainder:  3

Option 2: Synthetic Division 

Ex.  dividend/ divisor in form of (x-a)

Step 1: Arrange the coefficients of the dividend in order of descending powers of x (write 0 as the coefficient for each missing power) inside the division bracket. And write the divisor in the form of "x-a". Example:  x+3;  a= -3, since x-a = x -(-3) = x +3 . Then, place "a" outside the division bracket. 

 if a = -2, then x -(-a) = x-(-2) = x +2 [divisor]

Step 2: Bring down the first coefficient of the dividend and multiply it by "a", then add the product to the second coefficient of the dividend. Then repeat until the last term has a product.

* When using Synthetic Division, use ADDITION.

Step 3: The last number in the 3rd row is the remainder, while the other numbers are the coefficient of the quotient, which is a degree less than the dividend.

* Since the dividend has x^4, the quotient must have              x^3 (1 degree less)

   Quotient:  2x^3 -4x^2 +5x -6

Remainder:  3

Thank You and I hope that you have learned from this post. See Ya!