- Group like terms from both polynomials: \((2x^2 - 6x^2) + (8x - 5x) + (-7 + 6)\).
- Perform the arithmetic for each group: \(2 - 6 = -4\), \(8 - 5 = 3\), and \(-7 + 6 = -1\).
- Assemble the simplified terms into the final polynomial.
- The final expression is \(\boxed{-4x^2 + 3x - 1}\).
Explanation
- Analyzing the Polynomials using Algebra Tiles
To find the sum of the polynomials \((2x^2 + 8x - 7)\) and \((-6x^2 - 5x + 6)\), we treat the expressions as a collection of algebra tiles. Algebra tiles allow us to visualize polynomials by representing terms of the same degree with specific tiles: \(x^2\)-tiles for quadratic terms, \(x\)-tiles for linear terms, and unit tiles for constant terms. Positive terms are typically represented by one color, while negative terms are represented by another. Our goal is to combine the tiles of the same type and simplify the resulting collection.
- Grouping Like Terms
We group the terms of the same degree together from both polynomials:
- \(x^2\) terms: \(2x^2\) and \(-6x^2\)
- \(x\) terms: \(8x\) and \(-5x\)
- Constant terms: \(-7\) and \(6\)
The sum can be written as:
\[(2x^2 - 6x^2) + (8x - 5x) + (-7 + 6)\]
By grouping, we are essentially placing the corresponding algebra tiles side by side to count the net quantity of each type.
- Simplifying the Expression
Now, we simplify the coefficients for each degree:
- For the \(x^2\) terms, we combine \(2\) and \(-6\):
\[2 + (-6) = -4\]
This leaves us with \(-4x^2\).
- For the \(x\) terms, we combine \(8\) and \(-5\):
\[8 - 5 = 3\]
This leaves us with \(3x\).
- For the constants, we combine \(-7\) and \(6\):
\[-7 + 6 = -1\]
This leaves us with \(-1\).
As we combine the tiles, pairs of positive and negative tiles of the same type "cancel out" to zero, simplifying the expression to its minimal form.
- Conclusion
By combining all the simplified coefficients, we arrive at the final polynomial:
\[-4x^2 + 3x - 1\]
Thus, the sum of \((2x^2 + 8x - 7)\) and \((-6x^2 - 5x + 6)\) is \(\boxed{-4x^2 + 3x - 1}\).
Examples
Algebra tiles are a great way to visualize how algebraic expressions behave like physical objects. Imagine you are managing inventory for a store where you have boxes of different sizes: large boxes (\(x^2\)), medium boxes (\(x\)), and small items (units). If you have two different shipments and want to combine them, you group the same-sized boxes together and see how they total up. If one shipment includes returns (represented by negative numbers), those specific boxes cancel out the ones you already have. This simplifies your final inventory count, just as we simplified the polynomial expression \(-4x^2 + 3x - 1\).