To work with the given volume of the rectangular solid shipping container \( V(x) = 24x^3 + 6x^2 - 9x \), we can perform several operations such as factoring or analyzing the expression to understand more about its dimensions.
Step 1: Factor the volume expression
We can start by factoring the polynomial \( V(x) \).
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Factor out the greatest common factor (GCF): The GCF of the terms \( 24x^3, 6x^2, \) and \( -9x \) is \( 3x \). So, we factor out \( 3x \):
\[ V(x) = 3x(8x^2 + 2x - 3) \]
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Factor the quadratic \( 8x^2 + 2x - 3 \): We can factor the quadratic trinomial \( 8x^2 + 2x - 3 \) further. We look for two numbers that multiply to \( 8 \times -3 = -24 \) and add up to \( 2 \).
The pairs of factors of \( -24 \) that add to \( 2 \) are \( 6 \) and \( -4 \).
We can re-write the quadratic: \[ 8x^2 + 6x - 4x - 3 \]
Group the terms: \[ = (8x^2 + 6x) + (-4x - 3) \] \[ = 2x(4x + 3) - 1(4x + 3) \] \[ = (2x - 1)(4x + 3) \]
Final Factorization
Putting all of this together, we can express \( V(x) \) as: \[ V(x) = 3x(2x - 1)(4x + 3) \]
Step 2: Interpret the dimensions
Since \( V(x) \) represents the volume of a rectangular solid, we can interpret \( 3x, (2x - 1), \) and \( (4x + 3) \) as the three dimensions \( l, w, h \) of the rectangular solid:
- One dimension could be \( 3x \)
- The second could be \( 2x - 1 \)
- The third could be \( 4x + 3 \)
Step 3: Conditions for dimensions to be valid
For physical dimensions, the expressions for width and height must be positive:
- For \( 3x > 0 \): \( x > 0 \)
- For \( 2x - 1 > 0 \): \( x > \frac{1}{2} \)
- For \( 4x + 3 > 0 \): This is always true for \( x > 0 \) since \( 4x + 3 \) will always be positive.
Conclusion
The conditions determine that \( x \) needs to be greater than \( \frac{1}{2} \) for all dimensions to remain positive. The volume is factored as \( V(x) = 3x(2x - 1)(4x + 3) \), which gives us insight into the dimensions of the shipping container.