- Analyze the expression \(E = (p^3)(2p^2 - 4p)(3p^2 - 1)\).
- Apply the associative and distributive properties to expand \((2p^2 - 4p)(3p^2 - 1) = 6p^4 - 12p^3 - 2p^2 + 4p\).
- Compare the expanded form with the given options.
- The expression \((p^3)(6p^4 - 12p^3 - 2p^2 + 4p)\) matches option (b).
- Final Answer: \(\boxed{b}\).
Explanation
- Understanding the Problem and Properties
We are given the algebraic expression \(E = (p^3)(2p^2 - 4p)(3p^2 - 1)\). We need to determine which of the provided options is equivalent to this expression by applying the properties of multiplication, specifically the associative and distributive properties. Associative property of multiplication states that for any three expressions \(A, B, C\), the grouping of the multiplication does not change the result, i.e., \((A)(B)(C) = (A)(BC) = (AB)(C)\). The distributive property states that \(A(B + C) = AB + AC\).
- Testing the Options using Properties of Multiplication
Let us analyze the original expression \(E = (p^3)(2p^2 - 4p)(3p^2 - 1)\). We want to test which of the provided options is equal to \(E\).
Option b is \((p^3)(6p^4 - 12p^3 - 2p^2 + 4p)\).
By the associative property of multiplication, the expression \(E\) can be written as \((p^3) \cdot [(2p^2 - 4p)(3p^2 - 1)]\).
We can expand the term in the square brackets using the distributive property:
\[(2p^2 - 4p)(3p^2 - 1) = 2p^2(3p^2) + 2p^2(-1) - 4p(3p^2) - 4p(-1)\]
\[= 6p^4 - 2p^2 - 12p^3 + 4p\]
Rearranging the terms, we get:
\[6p^4 - 12p^3 - 2p^2 + 4p\]
Thus, the entire expression becomes:
\[(p^3)(6p^4 - 12p^3 - 2p^2 + 4p)\]
This is exactly the expression given in option b.
- Conclusion
We have shown that option b is equivalent to the original expression \(E\) by grouping the terms differently. The other options do not match the expansion:
- For option a, \((2p^6 - 4p^3)(3p^2 - 1)\) results in a higher degree polynomial when expanded.
- For option c, \((2p^5 - 4p^4)(3p^5 - p^3)\) involves terms with \(p^{10}\) and \(p^9\), which are not present in the original expression.
- For option d, \((p^3)(6p^4 + 4p) = 6p^7 + 4p^4\), which clearly differs from the original.
Therefore, option b is the correct equivalent expression. The final answer is \(\boxed{b}\).
Examples
Imagine you are an engineer optimizing the layout of a circuit or designing a software algorithm that performs calculations in stages. If you are calculating the total power dissipation or resource usage, you might arrive at a complex nested product, like \((p^3)(2p^2 - 4p)(3p^2 - 1)\). Knowing how to reorder or expand these terms allows you to compute the result in the most efficient manner, just as we did by grouping the terms differently to reach a simpler or more recognizable form, such as \((p^3)(6p^4 - 12p^3 - 2p^2 + 4p)\). This algebraic skill is essential for simplifying complex expressions in engineering, physics, and computer science.