Select the correct answer. Using synthetic division, what is the factored form of this polynomial? [tex]x^4+6 x^3+33 x^2+150 x+200[/tex]
A. [tex](x-2)(x-4)(x-5)(x+5)[/tex]
B. [tex](x+2)(x+4)(x-5)(x+5)[/tex]
C. [tex](x-2)(x-4)(x^2+25)[/tex]
D. [tex](x+2)(x+4)(x^2+25)[/tex]

Use synthetic division to test the roots from the given options.
Divide the polynomial by ( x + 2 ) using synthetic division, which results in x 3 + 4 x 2 + 25 x + 100 with no remainder.
Divide the result by ( x + 4 ) using synthetic division, which results in x 2 + 25 with no remainder.
The factored form of the polynomial is ( x + 2 ) ( x + 4 ) ( x 2 + 25 ) , so the correct answer is ( x + 2 ) ( x + 4 ) ( x 2 + 25 ) ​ .

Explanation

Problem Analysis We are given the polynomial x 4 + 6 x 3 + 33 x 2 + 150 x + 200 and asked to find its factored form using synthetic division. We are given four possible answers, and we will use synthetic division to check each one.

Testing Option D: Step 1 Let's test option D, ( x + 2 ) ( x + 4 ) ( x 2 + 25 ) . This means we can try dividing the polynomial by ( x + 2 ) and ( x + 4 ) using synthetic division. First, we divide by ( x + 2 ) , which means we use − 2 as the root. The coefficients of the polynomial are 1 , 6 , 33 , 150 , 200 .

Synthetic Division with -2 Using synthetic division with − 2 :






1
6
33
150
200



-2

-2
-8
-50
-200



1
4
25
100
0


The result is x 3 + 4 x 2 + 25 x + 100 with a remainder of 0. So, ( x + 2 ) is a factor.

Testing Option D: Step 2 Now, we divide the result x 3 + 4 x 2 + 25 x + 100 by ( x + 4 ) , which means we use − 4 as the root. The coefficients are 1 , 4 , 25 , 100 .

Synthetic Division with -4 Using synthetic division with − 4 :






1
4
25
100



-4

-4
0
-100



1
0
25
0


The result is x 2 + 0 x + 25 = x 2 + 25 with a remainder of 0. So, ( x + 4 ) is a factor.

Conclusion Therefore, the factored form of the polynomial is ( x + 2 ) ( x + 4 ) ( x 2 + 25 ) .

Final Answer Thus, the correct answer is D.


Examples
Factoring polynomials is a fundamental concept in algebra and has many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model market behavior, and computer scientists use it to design efficient algorithms. In structural engineering, understanding the roots of a polynomial can help determine the resonant frequencies of a bridge, ensuring it doesn't collapse under specific conditions. In economics, factoring can help identify equilibrium points in supply and demand models, providing insights into market dynamics. In computer science, factoring is used in cryptography to secure data transmission and storage.

Answered by GinnyAnswer | 2025-07-03

The polynomial x 4 + 6 x 3 + 33 x 2 + 150 x + 200 factors to ( x + 2 ) ( x + 4 ) ( x 2 + 25 ) using synthetic division. The correct answer is option D. Therefore, we can conclude that option D is the correct answer.
;

Answered by Anonymous | 2025-07-04