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 Len 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 3x³ + 9x² + 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 To determine the algebraic expressions for the length, width, and height of the rectangular prism, given the volume as 3x³ + 9x² + 6x, we need to factor this polynomial expression for volume and then assign the factors to the dimensions of the prism. Since the volume V is expressed as the product of length (l), width (w), and height (h), and the problem asks for algebraic representations, the key to this process is to factor the volume expression into a product of three algebraic terms.

First, observe the volume expression: 3x³ + 9x² + 6x. Notice that each term shares common factors. The coefficients 3, 9, and 6 have a greatest common factor of 3. Also, each term contains at least one factor of x. Therefore, the greatest common factor (GCF) of the entire expression is 3x. Factoring out 3x, we get: 3x(x² + 3x + 2) Next, focus on factoring the quadratic inside the parentheses: x² + 3x + 2. This is a quadratic trinomial, which can typically be factored into two binomials. To factor it, look for two numbers that multiply to 2 (the constant term) and add to 3 (the coefficient of the middle term). The numbers 1 and 2 satisfy these conditions because 1 × 2 = 2 and 1 + 2 = 3. Therefore, the quadratic factors into: (x + 1)(x + 2) Putting it all together, the factored form of the volume expression is: V = 3x(x + 1)(x + 2) Since the volume V is the product of length (l), width (w), and height (h), and given no specific ordering, we can assign these factors to the three dimensions in any order. Thus, the algebraic expressions for the dimensions are: Length = 3x

Width = x + 1 Height = x + 2 To verify, multiply these expressions: (3x) * (x + 1) * (x + 2) = 3x(x + 1)(x + 2) which matches the factored form of the volume, confirming the correctness of our assignment.

In summary, by factoring the given volume polynomial, we identified the algebraic expressions for the dimensions as 3x, x + 1, and x + 2. This algebraic approach clarifies how the dimensions relate to the variable x, providing a symbolic understanding of the prism's size in terms of algebraic expressions rather than numerical values.

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