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)

all group life insurance is term insurance. actually it is annual renewable term and it is rated on the average age and claims experience of the entire group, which is called "experience rating." remember, this type of insurance has a 31-day grace period and is convertible to whole life upon leaving employment. a contributory group plan requires that both the employer and the employees pay part of the premium. at least 75% of those eligible must participate, so the group is sure to get most of the healthy employees as well as those that are sick.

step-by-step explanation: