1.     When dividing polynomials, the degree of the quotient is equal to the difference between the degrees of the
       dividend and the divisor.

2.     Divisor) Dividend (quotient
                       ------------
                     Remainder

      Dividend = Divisor ∗ quotient + remainder

3.     If terms are missing in the dividend or divisor, leave spaces and treat them as terms with the coefficient zero.

We have observed that in the multiplication of polynomials, the degree of the product equals the sum of the
degrees of the multiplicand and the multiplier.

When dividing polynomials, the degree of the quotient is equal to the degree of the dividend minus the degree of
the divisor. The remainder may be zero or its degree is at least one less than that of the divisor.

Explaining Division of Polynomials:

1.     Divide the first term of the dividend by the first term of the divisor, i.e., x³ + x = x². This is the first term of the
       quotient.

      x³(x + 2) = x² + 2x²

       Write this under the dividend as shown, and subtract. We get a new dividend.




2.     The first term of the new dividend is x².

       x² + x = x

       Write this as the second term of the quotient.

       x(x + 2) = x² + 2x. Write this under the dividend and subtract. You get a new dividend again.




3.     The first term of the new dividend is -4x.

       -4x + x = -4. -4 is the third term of the quotient.

       -4(x + 2) = -4x -8

    Write this under the new dividend and subtract. 3 is left over.

    We stop the process as the remainder's degree is less than that of the divisor's.

   Quotient x² + x - 4, remainder 3.

    As in the case of natural numbers, the division algorithm, namely dividend = divisor quotient + remainder, can
   be used to verify our computations.

    Therefore,

    (x + 2) (x² + x - 4) + 3 = x³ + 3x² - 2x - 5.

If terms are missing in the dividend or divisor, leave spaces and treat them as terms with the coefficient zero.

The example explains the method of division of polynomials involving more than one variable by first arranging
the dividend and divisor in descending powers using one of the variables, and then dividing as follows.

Example:

Divide ( x³ + x² - 2x - 5 ) by ( x + 2 )

Here, both dividend and divisor are in descending powers of x. In the event that they are not so, write them in
descending powers of x.

                            Dividend

Divisor x + 2 ) x³ + 3x² - 2x - 5 (x² + x - 4 Quotient

                            x³ + 2x²

                            ______________

                                     x² - 2x

                                    x²+ 2x

                            ______________

                                             - 4x - 5

                                             - 4x - 8

                            ______________

                                                        3 Remainder.

Try these questions

1.     Divide ( x³- 5 x²+ 11x - 10) by ( x - 2 )

   Answer: Dividend

   Divisor x - 2) x³ - 5x²+ 11x - 10 (x² - 3x + 5 Quotient

                              - x³2x²

                               __________________

                                       -3x² + 11x

                                       3x² ± 6x

                               __________________

                                               5x - 10

                                               -5x 10

                               __________________

                                                           0 Remainder

   
   

   Verification:

   Dividend = Divisor ∗Quotient + Remainder

                     = ( x - 2 ) ∗( x² - 3x + 5 ) + 0

                     = x³ - 5 x² + 11x - 10

                     = Dividend.




2.    Divide ( - 2 x³ - 7 x²+ 8x + 5 ) by ( 2x - 3 )

   Answer: 2x - 3) 2x³ - 7x² + 8x + 5 (x² - 2x + 1

                              -2x³ 3x²

                               _________________

                                       - 4 x² + 8x

                                       4 x² ± 6x

                               _________________

                                               2x + 5

                                               -2x 3

                               _________________

                                                           8

   Verification:

   ( 2x -3 ) ( x² -2x + 1 ) + 8 = 2x ( x² -2x + 1 ) -3 ( x² -2x + 1 ) + 8

                     = 2x³ - 4x²+ 2x - 3x² + 3 + 6x - 3 + 8

                     = 2x³ - 7x² + 8x + 5

                     = Dividend.




3.     Divide ( x⁴ + 0x³ - 4 x² + 13x - 4 ) by ( x²- 2x + 3 )

   Answer: x² - 2x + 3) x⁴ + 0x³ - 4x² + 13x - 4 (x² + 2x - 3

                              - x⁴ 2x³ ± 3x²

                               _____________________

                                       2x³ - 7x² + 13x

                                       -2x³ 4x² ± 6x

                               ____________________

                                               - 3x² + 7x - 4

                                               3x² ± 6x 9

                               ____________________

                                                           x + 5

   
   

   Verification:

   ( x² - 2x + 3 ) ( x² + 2x - 3 ) + ( x + 5 )

                     = x⁴+ 2 x³ - 3 x² - 2 x³ - 4 x² + 6x + 3 x² + 6x - 9 + x + 5

                     = x⁴ - 4 x² + 13x - 4

                     = Dividend.




4.     Divide ( 3x⁴ - 8x³ + 10x²- 8x - 2 ) by ( 3x² - 2x + 5 )

   Answer:3x² – (2x + 5) 3x⁴ - 8x³ + 10x² - 8x - 2 (x² - 2x + 1/3

                              -3x⁴ 2x³ ± 5x²

                               ______________________

                                       - 6x³ + 5x² - 8x

                                        6x³ ± 4x² 10x

                               ______________________

                                                x² + 2x - 2

                                                x² (2/3)x ± 5/3

                               ______________________

                                                           (8/3)x -11/3

   
   

   Verification:

   ( 3x²- 2x + 5 ) ( x²- 2x + 1/3 ) + (8x)/3 - 11/3

                     = 3x⁴- 6x³+ x² - 2x³ + 4x²- (2x)/3 + 5x²- 10x + 5/3+ (8x)/3 - 11/3

                     = 3x⁴- 8x³+ 10x² - 8x - 2

                     = Dividend.




5.    Divide ( 4x⁴ - 8x³ + 9x² + 3x - 7 ) by ( 2 x² - x - 2 )

   Answer: 2x² - x - 2 ) 4x⁴ - 8x³ + 9x² + 3x - 7 (2x² - 3x + 5

                              -4x⁴ 2x³ 4x²

                               ______________________

                                       - 6x³ + 13 x² + 3x

                                       6x³ ± 3 x² ± 6x

                               ______________________

                                               10 x² - 3x - 7

                                               -10 x² 5x 10

                               ______________________

                                                           2x + 3

   
   

   Verification:

   ( 2x²- x - 2 ) ( 2x² - 3x + 5 ) + 2x + 3

                     = 4x⁴- 6x³ + 10x² - 2x³ + 3x²- 5x - 4x² + 6x -10 + 2x + 3

                     = 4x⁴ - 8x³+ 9x² + 3x - 7

                     = Dividend.




6.    Divide ( 8x⁴ - 8x³ - 10x² + 15x + 2 ) by ( 4x² + 2x - 3 )

   Answer: 4x² + 2x - 3) 8x⁴ - 8x³ - 10x² + 15x + 2 (2 x² - 3x + 1/2

                              8x⁴ + 4x³ - 6x²

                               ________________________

                                       - 12x³ - 4x²+ 15x

                                       -12x³ - 6x² + 9x

                               ________________________

                                                    2x² + 6x + 2

                                                    2x² + x - 3/2

                               ________________________

                                                                5x + 3 1/2

   
   

   Verification:

   ( 4x² + 2x - 3 ) ( 2x² - 3x + 1/2 ) + 5x + 7/2

                     = 8x⁴ - 12x³+ 2x² + 4x³ - 6x² + x - 6x² + 9x - 3/2 + 5x + 7/2

                     = 8x⁴ -8x ³- 10x² + 15x + 2

                     = Dividend.