Find the product.
$2 x(x-3)(x+2)$
A. $4 x-1$
B. $2 x^3-x-6$
C. $2 x^3-2 x^2-12 x$
D. $2 x^3-12 x$

First, multiply the binomials: ( x − 3 ) ( x + 2 ) = x 2 − x − 6 .
Then, multiply the result by 2 x : 2 x ( x 2 − x − 6 ) .
Distribute 2 x to each term: 2 x 3 − 2 x 2 − 12 x .
The final product is 2 x 3 − 2 x 2 − 12 x ​ .

Explanation

Understanding the Problem We are given the expression 2 x ( x − 3 ) ( x + 2 ) and asked to find the product. This involves expanding the expression by multiplying the terms together.

Multiplying the Binomials First, let's multiply the two binomials ( x − 3 ) and ( x + 2 ) :
( x − 3 ) ( x + 2 ) = x ( x + 2 ) − 3 ( x + 2 ) = x 2 + 2 x − 3 x − 6 = x 2 − x − 6

Multiplying by 2x Now, we multiply the result by 2 x :
2 x ( x 2 − x − 6 ) = 2 x ( x 2 ) − 2 x ( x ) − 2 x ( 6 ) = 2 x 3 − 2 x 2 − 12 x

Final Result Therefore, the product of the expression 2 x ( x − 3 ) ( x + 2 ) is 2 x 3 − 2 x 2 − 12 x .


Examples
Understanding polynomial multiplication is crucial in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution and ensure structural integrity. By expanding and simplifying these polynomial expressions, they can accurately predict the bridge's behavior under different conditions, ensuring safety and efficiency. This skill is also fundamental in computer graphics for rendering 3D models and creating realistic simulations.

Answered by GinnyAnswer | 2025-07-08