The Volume Of A Rectangular Prism Is Given By The Formula V

The Volume Of A Rectangular Prism Is Given By the Formula V Lwh The volume of a rectangular prism is given by the formula V = lwh (length•width•height). What are the length, width, and height of the prism if the volume is 3x 3 + 9x 2 + 6x? Your answer will not have exact numeric dimensions but rather three algebraic terms that represent them. To earn full credit, show how you solved the problem by using correct mathematical symbols and by explaining thoroughly in words each step.

Paper For Above instruction The problem asks us to determine the algebraic expressions representing the length, width, and height of a rectangular prism given its volume as 3x 3 + 9x 2 + 6x. Since the volume V of a rectangular prism is calculated by multiplying the length (l), width (w), and height (h), our task involves factoring the given polynomial volume expression into three algebraic factors corresponding to these dimensions. This process involves polynomial factorization, which helps us identify the individual dimensions as algebraic expressions.

Firstly, observe the volume expression: 3x 3 + 9x 2 + 6x. All terms share a common factor, which is 3x, since each term contains at least one factor of x and a numerical factor of 3. To factor out the greatest common factor (GCF), we proceed as follows:

Factor out 3x: 3x(x 2 + 3x + 2). This simplifies our problem to factoring the quadratic inside the parentheses: x 2 + 3x + 2.

Now, focus on factoring the quadratic expression: x 2 + 3x + 2. Looking for two numbers that multiply to 2 and add up to 3, we find these numbers are 1 and 2.

Therefore, the quadratic factors as: x 2 + 3x + 2 = (x + 1)(x + 2).

Putting it all together, the complete factorization of the volume polynomial is: V = 3x(x + 1)(x + 2).

Since the volume formula V = l × w × h corresponds to the product of three algebraic factors, an appropriate assignment is that the length, width, and height are represented by: l = 3x w=x+1 h=x+2 Alternatively, any permutation of these three expressions would equally satisfy the original volume expression, as the order of multiplication does not affect the volume. For example, the dimensions could also be (x + 1), (x + 2), and 3x in any order.

In conclusion, the algebraic expressions representing the length, width, and height of the rectangular prism are 3x, x + 1, and x + 2, respectively. This demonstrates how polynomial factorization enables us to identify the dimensions when given a volume expressed algebraically, illustrating the close relationship

between algebra and geometric concepts.

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