The volume of a rectangular prism box can be represented by the function [tex]$V(x)=2 x^3-5 x^2-3 x$[/tex]. If the height of the box is [tex]$x cm$[/tex], which of the following could represent the length and width of the container?
A. [tex]$2 x+1$[/tex] and [tex]$x+3$[/tex]
B. [tex]$2 x+1$[/tex] and [tex]$x-3$[/tex]
C. [tex]$2 x-1$[/tex] and [tex]$x+3$[/tex]
D. [tex]$2 x-1$[/tex] and [tex]$x-3$[/tex]
Answer (1)
The volume of a rectangular prism is given by V ( x ) = 2 x 3 − 5 x 2 − 3 x , and the height is x .
Factor out x : V ( x ) = x ( 2 x 2 − 5 x − 3 ) .
Factor the quadratic: 2 x 2 − 5 x − 3 = ( 2 x + 1 ) ( x − 3 ) .
The other two dimensions are 2 x + 1 and x − 3 .
Explanation
Understanding the Problem The problem states that the volume of a rectangular prism is given by the function V ( x ) = 2 x 3 − 5 x 2 − 3 x , and the height of the box is x cm. We need to find the expressions that represent the other two dimensions of the box. This means we need to factor the given volume function.
Factoring out the Height First, we can factor out the height x from the volume function: V ( x ) = x ( 2 x 2 − 5 x − 3 ) Now we need to factor the quadratic expression 2 x 2 − 5 x − 3 .
Factoring the Quadratic Expression To factor the quadratic expression 2 x 2 − 5 x − 3 , we look for two numbers that multiply to 2 × − 3 = − 6 and add up to − 5 . These numbers are − 6 and 1 . So we can rewrite the middle term as − 6 x + x :
2 x 2 − 5 x − 3 = 2 x 2 − 6 x + x − 3 Now we can factor by grouping: 2 x 2 − 6 x + x − 3 = 2 x ( x − 3 ) + 1 ( x − 3 ) = ( 2 x + 1 ) ( x − 3 ) So the factored form of the quadratic expression is ( 2 x + 1 ) ( x − 3 ) .
Finding the Other Dimensions Therefore, the volume function can be written as: V ( x ) = x ( 2 x + 1 ) ( x − 3 ) Since the height is x , the other two dimensions are ( 2 x + 1 ) and ( x − 3 ) .
Final Answer Comparing our result with the given options, we find that the correct answer is ( 2 x + 1 ) and ( x − 3 ) .
Examples
Understanding how to factor polynomials and apply them to geometric problems like finding the dimensions of a rectangular prism is useful in various real-world applications. For example, if you are designing a container with a specific volume and one dimension is constrained by some external factor, you can use polynomial factorization to determine the other dimensions. This is also applicable in fields like architecture and engineering where spatial arrangements and volume calculations are crucial.
