Question

Factor x^4 - 2x^3 - 33x^2 + 50x + 200 completely using the given factor x^2 - 25.

Answer 1

You can do synthetic division twice immediately (or once if you know how to handle x^2 - 25 all at once. I'll use 2 divisions. x^2 - 25 is (x + 5)(x - 5). Both of them are zeros to the given equation which means that x = +5 and

x = - 5  are both zeros of the given quartic.

5        1      -2       -33       50         200

                 5         15       -90         -200          

=====================================

          1       3        -18       -40         0

What you have now is x^3 + 3x^2 - 18x - 40 = 0

Do another synthetic division

-5        1       3       -18       -40

                   -5       10        40

============================

           1      -2        -8        0

The result is

x^2 - 2x - 8 = 0 which factors.

(x + 2)(x - 4) = 0

Answer

The factors are (x + 2)(x - 4)(x + 5)(x - 5) = 0

Long division

x^2 + 0 + 25 ||x^4 - 2x^3 - 33x^2 + 50x + 200 ||x^2 - 2x - 8

                      x^4 + 0  +    25x^2

                      ==================

                               -2x^3 -8x^2  +  50x

                                -2x^3 + 0  +     50x

                                =================

                                            -8x^2 + 0 +200

                                            -8x^2 + 0 + 200

                                ==================

                                                       0