- First product: \((x)(4x + 3) = \boxed{4x^2 + 3x}\)
- Second product: \((4x^2 + 3x)(4x) = \boxed{16x^3 + 12x^2}\)
- The property demonstrated is the \(\boxed{\text{commutative property}}\).
Explanation
- Calculating the first product
To find the expression for the volume of the box when multiplied in a specific order, we first calculate the product of the first two factors:
\[ (x)(4x + 3) \]
Distributing \(x\) across the binomial \((4x + 3)\), we have:
\[ x \cdot (4x) + x \cdot (3) = 4x^2 + 3x \]
So, \((x)(4x + 3) = 4x^2 + 3x\).
- Calculating the second product
Next, we take the result from the previous step and multiply it by the remaining factor \((4x)\):
\[ (4x^2 + 3x)(4x) \]
Distributing \(4x\) across the binomial \((4x^2 + 3x)\), we have:
\[ (4x^2) \cdot (4x) + (3x) \cdot (4x) = 16x^3 + 12x^2 \]
Thus, \((4x^2 + 3x)(4x) = 16x^3 + 12x^2\).
- Identifying the property of multiplication
The original expression was \(V = (x)(4x + 3)(4x)\). Since multiplication of polynomials is associative and commutative, changing the order of multiplication does not change the final product. The fact that the product remains consistent regardless of the order in which we multiply the factors demonstrates the commutative property of multiplication. Therefore, the property being illustrated is the commutative property.
Examples
In real-world manufacturing and engineering, calculating the volume of objects like boxes or containers is crucial for packaging design. For example, if you are designing a box to hold a specific product, you calculate the volume by multiplying length, width, and height. Because multiplication is commutative, it doesn't matter if you measure length then width then height, or height then length then width; the resulting volume capacity will be identical. Understanding these properties helps ensure consistent measurements and efficient material usage.