Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express P in the form P(x) = D(x) · Q(x) + R(x)

Question

P(x) = 64x^5 − 16x^4 + 4x^2 − 4, D(x) = 4x^2 − 4x + 1

Answer 1

First, we try to factor out D(x)

 

D(x) = (2x - 1)(2x - 1)

 

 

Since D(x) is a factor, 1/2 could be a root.  We can use synthetic division to find this out.

 

1/2 | 64       -16       0         4        0        -4

 

                    32       8         4        4         2

_______________________________________

        64        16        8         8        4        -2

 

 

1/2 is not a root because the remainder is not zero.  Therefore, this quotient cannot be used.   So we must use long division.

 

                                   

                                            16x³ + 12x² + 8x + 6

                      _______________________________

(4x² - 4x + 1) | 64x⁵ - 16x⁴ + 0x³ + 4x² + 0x - 4

                        64x⁵ - 64x⁴ +16x³

                        _________________

                                 48x⁴ - 16x³ + 4x²

                                 48x⁴ - 48x³ +12x²

                                 ___________________

                                           32x³ - 8x² + 0x

                                           32x³ - 32x² +8x

                                           _______________

                                                    24x² - 8x    - 4

                                                    24x² - 24x + 6

                                                    ____________

                                                             16x - 10

 

Q(x) = 16x³ + 12x² + 8x + 6

R(x) = 16x - 10