- Distribute the negative sign in each expression to remove parentheses.
- Group the \(x^2\) terms and constant terms separately.
- Simplify by performing the arithmetic operations on coefficients and constants.
- Identify that \((10x^2 - 4) - (7x^2 + 3) = 10x^2 - 4 - 7x^2 - 3 = 3x^2 - 7\).
- The correct expression is \((10x^2 - 4) - (7x^2 + 3)\) which corresponds to \(\boxed{(10x^2 - 4) - (7x^2 + 3)}\).
Explanation
- Analysis of the Problem
To determine which expression is equivalent to \((3x^2 - 7)\), we evaluate each of the four given options by performing the polynomial subtraction. Algebraic equivalence means that after distributing the negative sign across the second polynomial in each expression and combining like terms, the result must match the original expression.
- Simplifying Each Candidate Expression
We simplify each candidate expression one by one:
- Choice 1: \((2x^2 - 11) - (x^2 + 4) = 2x^2 - 11 - x^2 - 4\). Grouping like terms: \((2x^2 - x^2) + (-11 - 4) = x^2 - 15\).
- Choice 2: \((5x^2 - 6) - (2x^2 - 1) = 5x^2 - 6 - 2x^2 + 1\). Grouping like terms: \((5x^2 - 2x^2) + (-6 + 1) = 3x^2 - 5\).
- Choice 3: \((10x^2 - 4) - (7x^2 + 3) = 10x^2 - 4 - 7x^2 - 3\). Grouping like terms: \((10x^2 - 7x^2) + (-4 - 3) = 3x^2 - 7\).
- Choice 4: \((15x^2 + 8) - (18x^2 + 1) = 15x^2 + 8 - 18x^2 - 1\). Grouping like terms: \((15x^2 - 18x^2) + (8 - 1) = -3x^2 + 7\).
- Conclusion and Final Result
Comparing the simplified results with the target expression \((3x^2 - 7)\):
- Choice 1 simplifies to \(x^2 - 15\).
- Choice 2 simplifies to \(3x^2 - 5\).
- Choice 3 simplifies to \(3x^2 - 7\).
- Choice 4 simplifies to \(-3x^2 + 7\).
Only the third expression, \((10x^2 - 4) - (7x^2 + 3)\), results in the equivalent form \((3x^2 - 7)\). Therefore, the correct choice is the third one.
Examples
In real-world applications like financial modeling, you often need to simplify complex equations describing revenue and cost structures. For instance, if you represent your total profit as a function of production units, say \(P(x) = 3x^2 - 7\), you might find that different accounting methods yield different initial expressions, such as \((10x^2 - 4) - (7x^2 + 3)\). Learning to simplify these polynomials ensures that different formulas are correctly reconciled to the same underlying economic model, allowing for accurate predictions and decision-making.