- Expand the terms within the polynomial expression by distributing the multiplication and the negative sign.
- Combine the like terms related to \(v^4\), \(v^4 w^2\), \(v^2 w^3\), and the constants.
- Verify the simplification against the provided options.
- The equivalent expression is \(\boxed{v^4 w^2+15 v^2 w^3+2}\).
Explanation
- Expanding the expression terms
To simplify the polynomial expression \((9 v^4+2)+v^2(v^2 w^2+2 w^3-2 v^2)-(-13 v^2 w^3+7 v^4)\), we first expand the terms inside the parentheses.
- The first part is \((9v^4 + 2)\).
- For the second part, distribute \(v^2\) across \((v^2 w^2+2 w^3-2 v^2)\):
\[v^2(v^2 w^2) + v^2(2 w^3) - v^2(2 v^2) = v^4 w^2 + 2 v^2 w^3 - 2 v^4\]
- For the third part, distribute the negative sign across \((-13 v^2 w^3+7 v^4)\):
\[-(-13 v^2 w^3) - (7 v^4) = 13 v^2 w^3 - 7 v^4\]
- Combining like terms and simplifying
Now, combine all the expanded parts into a single expression:
\[(9 v^4 + 2) + (v^4 w^2 + 2 v^2 w^3 - 2 v^4) + (13 v^2 w^3 - 7 v^4)\]
Group the like terms:
- For \(v^4\): \(9 v^4 - 2 v^4 - 7 v^4 = (9 - 2 - 7) v^4 = 0 v^4 = 0\)
- For \(v^4 w^2\): Only one term, which is \(v^4 w^2\)
- For \(v^2 w^3\): \(2 v^2 w^3 + 13 v^2 w^3 = 15 v^2 w^3\)
- Constant term: \(2\)
Combining these results, we get the simplified polynomial:
\[v^4 w^2 + 15 v^2 w^3 + 2\]
- Final conclusion
Comparing our simplified result \(v^4 w^2 + 15 v^2 w^3 + 2\) with the given options:
- Option 1: \(v^4 w^2+15 v^2 w^3+2\)
- Option 2: \(14 v^4+v^4 w^2+15 v^2 w^3+2\)
- Option 3: \(16 v^6 w^5+2\)
- Option 4: \(14 v^4+16 v^6 w^5+2\)
The simplified expression matches the first option. Therefore, the correct choice is the first one. The final result is \(\boxed{v^4 w^2+15 v^2 w^3+2}\).
Examples
In financial modeling or engineering, expressions like this often represent the cost of materials or resource allocation across different variables (like volume \(v\) and width \(w\)). By simplifying complex formulas into their lowest terms, we can quickly calculate final costs or values, making decision-making much more efficient and less prone to calculation errors. For example, if \(v\) and \(w\) were dimensions, this simplification would help calculate the surface area or volume of a complex geometric shape in one clean step.