The long division below shows the first term of the quotient. Which polynomial should be subtracted from the dividend first?

$x + 2 \longdiv { x ^ { 3 } + 3 x ^ { 2 } + x }$

A. $x+3$
B. $x^2+x+2$
C. $x^3+2 x^2$
D. $x^2+3 x$

Multiply the first term of the quotient ( x 2 ) by the divisor ( x + 2 ).
Calculate the product: x 2 ( x + 2 ) = x 3 + 2 x 2 .
The polynomial to be subtracted from the dividend first is x 3 + 2 x 2 .
The final answer is x 3 + 2 x 2 ​ .

Explanation

Understanding the Problem We are given a long division problem and asked to determine the polynomial that should be subtracted from the dividend in the first step. The divisor is x + 2 and the dividend is x 3 + 3 x 2 + x . The first term of the quotient is implicitly given as x 2 .

Finding the Polynomial to Subtract To find the polynomial to subtract, we multiply the first term of the quotient, x 2 , by the divisor, x + 2 .

Calculating the Product Calculating the product: x 2 ( x + 2 ) = x 2 ⋅ x + x 2 ⋅ 2 = x 3 + 2 x 2 .

Conclusion Therefore, the polynomial that should be subtracted from the dividend first is x 3 + 2 x 2 .


Examples
Long division of polynomials is used in various engineering and scientific applications, such as control systems design, signal processing, and cryptography. For example, in control systems, polynomial division can help simplify transfer functions, making it easier to analyze and design controllers. In signal processing, it can be used to decompose signals into simpler components. Understanding polynomial division is also crucial in cryptography for tasks like error correction and data encryption.

Answered by GinnyAnswer | 2025-07-05