- Analyze the expression to identify like terms: \(x^2\) terms, \(x\) terms, and constant terms.
- Group the like terms: \((x^2 - 10x^2) + (-x) + (10 + 8)\).
- Perform the arithmetic for each group to get \(-9x^2\), \(-x\), and \(18\).
- Final simplified expression: \(\boxed{-9x^2 - x + 18}\).
Explanation
- Problem Analysis
The objective is to simplify the algebraic expression \((x^2+10)+(-10x^2-x+8)\) by combining like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two quadratic polynomials being added together. The first is \(x^2 + 10\) and the second is \(-10x^2 - x + 8\). Identifying the terms in each polynomial allows us to group them according to their variable powers.
- Grouping Like Terms
To simplify the addition, we look for terms that contain the same power of \(x\).
- The quadratic terms are \(x^2\) and \(-10x^2\).
- The linear terms are \(0x\) (as there is no \(x\) term in the first polynomial) and \(-x\).
- The constant terms are \(10\) and \(8\).
By grouping these, we rewrite the expression as \((x^2 - 10x^2) + (-x) + (10 + 8)\).
- Combining the Terms
Now, we perform the addition for each group:
- Combining the quadratic terms: \(x^2 - 10x^2 = (1 - 10)x^2 = -9x^2\).
- The linear term remains as \(-x\).
- Combining the constant terms: \(10 + 8 = 18\).
Adding these results together gives the final simplified expression.
- Final Result
Putting the terms together, the result of the simplification is \(-9x^2 - x + 18\). Thus, \((x^2+10)+(-10x^2-x+8) = \boxed{-9x^2 - x + 18}\).
Examples
In real-world inventory management, imagine you have two warehouses tracking similar products. Warehouse A holds \(10\) items and a certain quantity of base units represented by \(x^2\), while Warehouse B holds \(8\) additional items, loses \(x\) units of a specific resource, and has \(-10\) times the amount of base units in terms of debt or stock depletion. Combining these inventories to find the net balance involves exactly the same algebraic process of grouping like terms to reach a final total, ensuring you have an accurate count of all resources.