Using synthetic division, what is the factored form of this polynomial?

option D

Step-by-step explanation:

Since no factors are mentioned at all. We use trial and error method to find a root of  the polynomial.

Step 1 :

For that we will consider the factors of 200 :

± 1 , ±2, ± 4, ±5, ±8, ±10, ±20, ±25, ±40, ±50, ±100, ±200

f ( -1 ) = (-1)⁴ + 6 (-1)³ + 33(-1)² + 150(-1) + 200

      = 1 - 6 + 33 - 150 + 200

      = 78 ≠ 0

So - 1 is not a root.

f ( - 2) = ( -2)⁴ + 6( - 2 )³ + 33 (- 2) ² + 150 (- 2) + 200

        = 16 - 48 + 132 - 300 + 200

       = 0

Therefore - 2 is a root of f( x ) and ( x + 2) is a factor of f( x) .

Step 2

Now we will use synthetic division to find other factors.

    -2 | 1          6          33        150        200

        | 0        -2          -8         -50        -200

        |_______________________________

         1          4          25          100          0

We will use trial and error method again to further factorize the equation

g(x) = x³ + 4x² + 25x +100.

Proceed as in step 1

Consider the factors of 100 to  find the root.

Factors of 100 :±1, ±2, ±4 , ±5 , ±8, ±10 , ± 20, ±25, ± 50, ±100

g( - 4) = ( -4)³ + 4 ( -4)² + 25(-4) + 100

        = - 64 + 64 - 100 + 100

        = 0

Therefore - 4 is a root of  g(x) , that means (x + 4) is a factor.

Now we will use synthetic division :

     - 4 | 1         4        25        100

          | 0       -4         0        -100

          |_____________________

            1         0        25          0

Therefore g(x) = ( x² + 0x + 25)(x + 4)

                      = ( x² + 25 )( x + 4)

Therefore the factorized form of

x⁴ + 6x³  + 33x² + 150x + 200 = (x + 2)(x³ + 4x² + 25x + 100)

                                              = ( x + 2 )( x + 4 )( x² + 25)