- Identify like terms in both polynomials: \(x^2\) terms, \(x\) terms, and constants.
- Combine the coefficients of each set of like terms: \((8-5)x^2\), \((6-5)x\), and \((-2+5)\).
- Simplify the resulting expression to get \(\boxed{3x^2 + x + 3}\).
Explanation
- Identifying and Grouping Like Terms
To add the polynomials \((8x^2 + 6x - 2)\) and \((-5x^2 - 5x + 5)\), we identify and combine the like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have \(x^2\)-terms, \(x\)-terms, and constant terms. Using the concept of algebra tiles, we can think of these as:
- \(8\) positive \(x^2\)-tiles and \(5\) negative \(x^2\)-tiles.
- \(6\) positive \(x\)-tiles and \(5\) negative \(x\)-tiles.
- \(2\) negative unit tiles and \(5\) positive unit tiles.
- Combining Like Terms
Next, we combine the like terms by adding their coefficients:
- For the \(x^2\)-terms: \(8x^2 + (-5x^2) = (8 - 5)x^2 = 3x^2\).
- For the \(x\)-terms: \(6x + (-5x) = (6 - 5)x = 1x = x\).
- For the constant terms: \(-2 + 5 = 3\).
By combining these results, we get the simplified polynomial.
- Final Result
By grouping the results from the previous step, the sum of the polynomials \((8x^2 + 6x - 2) + (-5x^2 - 5x + 5)\) is \(3x^2 + x + 3\). This represents the final simplified form of the expression after canceling out opposite-colored tiles (a positive tile and a negative tile summing to zero). Thus, the result is \(\boxed{3x^2 + x + 3}\).
Examples
Algebra tiles are a great visual tool for understanding how expressions work, similar to balancing a budget or managing resources. Imagine you are tracking inventory for a store: if you have 8 units of product A and 5 go out, you are left with 3. If you do this for different product types simultaneously, you are effectively performing polynomial addition. Just as in algebra, keeping track of separate categories (like different products) ensures you have a clear picture of your total remaining assets.