Question

select the correct answer.
using synthetic division, what is the factored form of this polynomial?
$x^4 + 6x^3 + 33x^2 + 150x + 200$
a. $(x + 2)(x + 4)(x - 5)(x + 5)$
b. $(x + 2)(x + 4)(x^2 + 25)$
c. $(x - 2)(x - 4)(x^2 + 25)$
d. $(x - 2)(x - 4)(x - 5)(x + 5)$

Explanation:

Step1: Test root x=-2 via synthetic division

Set up synthetic division for $x^4 + 6x^3 + 33x^2 + 150x + 200$ with root -2:

$$\begin{array}{r|rrrrr} -2 & 1 & 6 & 33 & 150 & 200 \\ & & -2 & -8 & -50 & -200 \\ \hline & 1 & 4 & 25 & 100 & 0 \end{array}$$

Resulting polynomial: $x^3 + 4x^2 + 25x + 100$

Step2: Factor the cubic polynomial

Factor by grouping:
$x^3 + 4x^2 + 25x + 100 = x^2(x+4) + 25(x+4) = (x+4)(x^2+25)$

Step3: Combine all factors

Original polynomial factors as $(x+2)(x+4)(x^2+25)$

Step4: Verify against options

Match with option B, confirm no further real roots ($x^2+25$ has no real factors).

Answer:

B. $(x + 2)(x + 4)(x^2 + 25)$